## Lesson

Let's investigate expressions with variables and exponents.

Exercise (PageIndex{1}): Up or Down?

Find the values of (3^{x}) and (left(frac{1}{3} ight)^{x}) for different values of (x). What patterns do you notice?

(x) | (3^{x}) | (left(frac{1}{3}
ight)^{x}) |
---|---|---|

(1) | ||

(2) | ||

(3) | ||

(4) |

Exercise (PageIndex{2}): What's the Value?

Evaluate each expression for the given value of (x).

- (3x^{2}) when (x) is (10)
- (3x^{2}) when (x) is (frac{1}{9})
- (frac{x^{3}}{4}) when (x) is (4)
- (frac{x^{3}}{4}) when (x) is (frac{1}{2})
- (9+x^{7}) when (x) is (1)
- (9+x^{7}) when (x) is (frac{1}{2})

Exercise (PageIndex{3}): Exponent Experimentation

Find a solution to each equation in the list. (Numbers in the list may be a solution to more than one equation, and not all numbers in the list will be used.)

- (64=x^{2})
- (64=x^{3})
- (2^{x}=32)
- (x=left(frac{2}{5} ight)^{3})
- (frac{16}{9}=x^{2})
- (2cdot 2^{5}=2^{x})
- (2x=2^{4})
- (4^{3}=8^{x})

List:

(frac{8}{125}quad frac{6}{15}quadfrac{5}{8}quadfrac{8}{9}quad 1frac{4}{3}quad 2quad 3quad 4quad 5quad 6quad 8)

Are you ready for more?

This fractal is called a Sierpinski Tetrahedron. A tetrahedron is a polyhedron that has four faces. (The plural of tetrahedron is tetrahedra.)

The small tetrahedra form four medium-sized tetrahedra: blue, red, yellow, and green. The medium-sized tetrahedra form one large tetrahedron.

- How many small faces does this fractal have? Be sure to include faces you can’t see. Try to find a way to figure this out so that you don’t have to count every face.
- How many small tetrahedra are in the bottom layer, touching the table?
- To make an even bigger version of this fractal, you could take four fractals like the one pictured and put them together. Explain where you would attach the fractals to make a bigger tetrahedron.
- How many small faces would this bigger fractal have? How many small tetrahedra would be in the bottom layer?
- What other patterns can you find?

### Summary

In this lesson, we saw expressions that used the letter (x) as a variable. We evaluated these expressions for different values of (x).

- To evaluate the expression (2x^{3}) when (x) is (5), we replace the letter (x) with (5) to get (2cdot 5^{3}). This is equal to (2cdot 125) or just (250). So the value of (2x^{3}) is (250) when (x) is (5).
- To evaluate (frac{x^{2}}{8}) when (x) is (4), we replace the letter (x) with (4) to get (frac{4^{2}}{8}=frac{16}{8}), which equals (2). So (frac{x^{2}}{8}) has a value of (2) when (x) is (4).

We also saw equations with the variable (x) and had to decide what value of (x) would make the equation true.

- Suppose we have an equation (10cdot 3^{x}=90) and a list of possible solutions: (1, 2, 3, 9, 11). The only value of (x) that makes the equation true is (2) because (10cdot 3^{2}=10cdot 3cdot 3), which equals (90). So (2) is the solution to the equation.

## Practice

Exercise (PageIndex{4})

Evaluate each expression if (x=3).

- (2^{x})
- (x^{2})
- (1^{x})
- (x^{1})
- (left(frac{1}{2} ight)^{x})

Exercise (PageIndex{5})

Evaluate each expression for the given value of each variable.

- (2+x^{3}), (x) is (3)
- (x^{2}), (x) is (frac{1}{2})
- (3x^{2}+y), (x) is (5) (y) is (3)
- (10y+x^{2}), (x) is (6) (y) is (4)

Exercise (PageIndex{6})

Decide if the expressions have the same value. If not, determine which expression has the larger value.

- (2^{3}) and (3^{2})
- (1^{31}) and (31^{1})
- (4^{2}) and (2^{4})
- (left(frac{1}{2} ight)^{3}) and (left(frac{1}{3} ight)^{2})

Exercise (PageIndex{7})

Match each equation to its solution.

- (7+x^{2}=16)
- (5-x^{2}=1)
- (2cdot 2^{3}=2^{x})
- (frac{3^{4}}{3^{x}}=27)

- (x=1)
- (x=2)
- (x=3)
- (x=4)

Exercise (PageIndex{8})

An adult pass at the amusement park costs 1.6 times as much as a child’s pass.

- How many dollars does an adult pass cost if a child’s pass costs:

($5)?

($10)?

(w) dollars? - A child's pass costs $15. How many dollars does an adult pass cost?

(From Unit 6.2.1)

Exercise (PageIndex{9})

Jada reads 5 pages every 20 minutes. At this rate, how many pages can she read in 1 hour?

- Use a double number line to find the answer.

- Use a table to find the answer.

pages read | time in minutes |
---|---|

(5) | (20) |

Which strategy do you think is better, and why?

(From Unit 2.4.4)

## Lesson 6

Most fractional exponents students have encountered so far in the unit are unit fractions. This warm-up allows students to recall how to interpret exponents that are non-unit fractions. Students are given a set of expressions that are all equivalent and asked to explain why they are so. In doing so, they practice applying properties of exponents and recall how to make sense of an expression such as (8 ^

- Discuss with a partner why the following expressions are equivalent.
- (64^frac13)
- (left(8^2 ight)^frac13)
- (left(2^3 ight)^frac23)
- (left(8^frac13 ight)^2)
- (8^frac23)

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

Select students to share their explanations as to why each expression is equivalent to the others. Time permitting, after a student shares their reasoning for a particular expression, ask if others thought about the same expression differently.

If not mentioned in students’ explanations, highlight that each of the expressions can be shown to be equivalent to (8^frac23) . Also point out that all expressions are equivalent because when each is raised to the third power, it is equal to (8^2) .

## An Introduction to Dynamics of Colloids

### The effect of translational and rotational coupling

An analysis as given above with the inclusion of the coupling term is a complicated matter. Fortunately, the relevance of the coupling term can be analysed, quite straightforwardly, in the short-time limit. To this end we employ the operator exponential expression (1.67) for correlation functions that is derived in the introductory chapter, in the same way as for the derivation of short-time results for a system of interacting spheres in section 6.5 . Since the stochastic variable here is X = ( r , u ˆ ) , and the pdf P ( X ) = 1 4 π V , we have,

In the second equation, the operator exponential is Taylor expanded up to first order in time. Only the coupling term in the Smoluchowski operator (the term ∼ Δ D ) needs be considered here. Substitution of that term into the above expression leads to (with

*x*= cos<θ>),Exponentiation of the resulting short-time expression gives rise to an additional factor exp < − c ( k L ) Δ D k 2 t >to the EACF,

The

*coupling function C*(*kL*) is equal to,This function can be evaluated numerically and is plotted in fig.6.17 . Since the decay time of the EACF due to translation is typically 1 / D ¯ k 2 , and Δ

*D*is not much different from D ¯ (see eqs. ( 5.127 , 128 )), the additional contribution due to the coupling of translation and rotation may be neglected for*kL*< 5, as can be seen from fig.6.17 (there is a 10% contribution from coupling at times where D ¯*k*2*t*≈ 1). Since we found earlier that a significant contribution from rotational diffusion is only found for*kL*> 5, the conclusion is that coupling between translation and rotation is of importance as soon as rotational diffusion itself is of importance.*Rotational diffusion and the coupling with translational diffusion are both important (for kL*> 5))*or essentially absent (for kL*< 5).Figure 6.17 .

*The coupling function C, which characterizes the relevance of coupling of translational and rotational diffusion, as a function of kL*.There are thus three regions in wavevector space to be distinguished. For

*kL*< 5 only translational diffusion is observed, while for 5 <*kL*< 10 both translational and rotational diffusion are observed, including the coupling between translation and rotation, and where the EACF (6.221) may be truncated after the*S*_{2}-term. For*kL*> 10, higher order rotational terms in the expansion (6.221) become relevant in addition (for 10 <*kL*< 15, the only significant higher order contribution is the*S*_{4}-term).The EACF is a sum of two exponential functions of time for

*kL*< 10. The difference with the case where coupling is neglected, is that D ¯*k*2*t*is now replaced by [*C*(*kL*)Δ*D*+ D ¯ ]*k*2*t*. The EACF is now a sum of two exponential functions in time, with a frequency Γ 1 = [ c ( k L ) Δ D + D ¯ ] k 2 , and a frequency Γ 2 = [ c ( k L ) Δ D + D ¯ ] k 2 + 6 D T · The amplitude of the latter exponential function relative to the first (which ratio is equal to*S*_{2}(*kL*)/*S*_{0}(*kL*)) is significantly different from zero only for*kL*> 5. The experimental procedure to measure both D ¯ and*D*is thus as follows. First perform measurements of the IACF for wavevectors_{r}*kL*< 5, and fit the IACF via the Siegert relation (3.82) with g ˆ E = exp < − Γ t >· In this wavevector range, Γ = D ¯ k 2 , since both rotation and coupling are insignificant. Determine the average translational diffusion coefficient D ¯ from the slope of a plot of Γ versus*k*2 . Then subtract the two frequencies, Γ_{2}− Γ_{1}= 6*D*, as obtained from measurements in the wavevector range 5 <_{r}*kL*< 10. This difference should be wavevector independent, and gives a value for the rotational diffusion coefficient*D*._{r}

## 6.3.4: Equivalent Exponential Expressions - Mathematics

o Derive the rules for multiplying and dividing exponential expressions

o Determine the meaning of a negative exponent

o Apply exponents to understanding and using scientific notation

Let's say we want to multiply two exponential expressions with the same base, such as and . The "brute force" approach to finding the product would be to expand each exponent, multiply the results, and convert back to an exponent (assuming an exponential representation of the result is desired).

Note carefully that when we multiply two exponents (again, assuming they have the same base), the result is multiplication of the factors of the first exponent and the factors of the second exponent. The total number of factors is thus the sum of the two exponents. We can generalize this rule using letters to stand in the place of unspecified numbers.

You may often see the multiplication operation expressed using a dot (·) instead of a cross ( ), or you may see it expressed without any symbol at all. Thus, each of the following expressions is equivalent to the others.

We can derive a similar rule for division. Let's take a look at what happens when we divide by .

Now, we can "cancel" any instance of a factor that appears in both the numerator and denominator. Why is this the case? Recall that we were able to find equivalent fractions by multiplying (or dividing) both the numerator and denominator by a particular value-this is equivalent to multiplying or dividing by one. Thus, we can do the following:

This is nothing more than writing the original fraction in an equivalent form. Instead of the fraction involving a single number, it involves a series of operations (multiplication, in this case).

The simple way to look at this is that any factors in the numerator can simply cancel equivalent factors in the denominator. Thus, for every instance where 2 appears in the numerator and denominator, we can cross that pair off.

We can see that the exponent of the answer is the difference between that of the numerator and that of the denominator (again, all have the same base). Let's generalize the rule:

Let's consider one more case: what if an exponential expression is itself raised to an exponent, as with the example below?

Expand the expression to see what it looks like in terms of multiplication.

Notice that the expression in parentheses has three factors, and we must multiply this expression four times. Thus, the total number of factors of two is 12, or the product of the exponents. Generally, the rule can be stated as follows.

The key to using these rules is to note that the exponential expressions must always have the same base-the rules do not apply to exponents with different bases. To recap, the rules of exponents are the following.

Write each of the following as an exponential expression with a single base and a single exponent.__Practice Problem__:In each case, use the rules for multiplying and dividing exponents to simplify the expression into a single base and a single exponent. Note in part c that any number is equal to itself raised to the first power. Note also the usefulness of these rules of exponents in part d: multiplying 150 twenty (or twenty-six) times is a tough proposition, and even most calculators cannot provide an exact product. The use of exponents, however, allows us to have an__Solution__:*exact*representation of the result.We now look at a slightly different case: a product of two (or more) factors, all raised to an exponent. Let's consider the example below.

Expand the exponential in the usual manner:

Notice that the expanded form has four factors of 2 and four factors of 3. This expression is the same as the following (multiplication is commutative, so we can rearrange the factors in the product).

Thus, given the product

*ab*(or ), we can write the following general rule:If you consider division of exponential expressions, you may notice that the rule seems to indicate that we can end up with negative exponents. The following example, similar to an example above, illustrates this point:

But the rules of exponents indicate the following:

The results

*should*be the same so, let's equate the two and see what information we can glean from the result.Let's rewrite the exponential expression as follows.

But the rules of exponents allow us to write this expression as an exponent raised to another exponent.

This result implies that an exponent of is associated with the reciprocal:

Thus, any number

*a*raised to the power of is equal to .Using our rules of exponents, therefore, we can determine generally what it means to raise a number to a power that is a negative number (specifically at this point, a negative integer). The derivation below follows the pattern of the example we considered above.

This result can be combined with the result we obtained previously for a product raised to a given power:

Let's consider a quotient (fraction) raised to a power. We can use the exponent rules we have studied thus far to derive an equivalent result. First, we'll use the above product rule as shown below.

Now, we'll use a negative exponent and other exponent rules.

Remember that multiplication is commutative. Thus,

Rewrite the second exponential as follows.

Thus, we obtain a rule for quotients similar to that for products:

Thus, we are now able to handle any integer exponents, whether positive or negative. We also know how to multiply and divide exponential expressions. Later in the course, we will consider fractional exponents. (As it turns out, fractional exponents obey the same rules as integer exponents, but the precise meaning of a fraction will be made clear later on.)

Evaluate or simplify each expression.__Practice Problem__:Use the rules of exponents along with the fact that a negative exponent indicates that the reciprocal of the base must be taken.__Solution__:*Application of Exponents: Scientific Notation*Although exponents may at times seem like an obscure or less than practical mathematical tool, they have numerous important and practical applications. For instance, exponents are used in so-called

**scientific notation,**which is a way of representing decimal values that are very large or very small. Consider the two numbers below.Because of their sizes, writing these numbers is extremely cumbersome. If we could write them simply as 1.37 followed by some indication of the number of zeroes that follow or precede these digits, we could make the process of expressing these numbers much simpler. Note that we can obtain the larger number by repeatedly multiplying 1.37 by 10 some number of times.

The expression on the right must include 26 factors of 10. But we know how to write multiple factors using exponents:

We can do likewise with the smaller number. In this case, 1.37 must be divided by 10 some number of times.

In this case, the denominator contains 22 factors of 10. Let's use what we know about negative exponents to write this in a form similar to the one we used above for the larger number.

Scientific notation is this method of representing numbers. The general format is a single integer digit followed by some number of decimal places, all multiplied by an integer power of 10. Let's take a look at two more examples of conversion from standard notation to scientific notation.

Note that the exponent can also be viewed as the number of places that the decimal point must be moved to the left (for positive exponents) or to the right (for negative exponents) when going from the standard number to scientific notation.

Convert each number to scientific notation.__Practice Problem__:a. 0.0000041 b. ,720,000 c. 0.0839

For each number, write a decimal containing the non-zero digits with the decimal point following the first digit, then multiply by the appropriate power of 10.__Solution__:Convert each number in scientific notation to a decimal in standard form.__Practice Problem__:Conversion to standard form simply requires movement of the decimal point the number of places indicated by the exponent. For a negative exponent, the decimal point must be moved to the left, and for a positive exponent, it must be moved to the right.__Solution__:

## Behold the Power of Exponents

You may have seen teeny little numbers floating above and to the right of other numbers and variables, like in the expression

*x*3 . What is that little 3 doing up there? Parasailing? Is it small because of scale, perhaps because it is actually as large as the earth's sun in reality, but is only seen from hundreds of thousands of miles away? No, that little guy is called the*exponent*or*power*of*x*in the expression, and indeed it has a very*power*ful role in algebra.##### Talk the Talk

In the exponential expression

*y*4 , 4 is the**exponent**and*y*is the**base**.### Big Things Come in Small Packages

The role of an exponent is to save you time and to clean up the way expressions are written. Basically, an exponent is a shorthand way to indicate repeated multiplication.

In the language of algebra,

*x*3 (read "*x*to the third power") means "*x*multiplied by itself three times", or*x**x**x*. To find the value of real numbers raised to exponents, just multiply the large number attached to the exponent (called the*base*) by itself the indicated number of times.**Example 2**: Evaluate the exponential expressions.- (a) 4 3
**Solution**: In this expression, 4 is the base and 3 is the exponent. To find the answer, multiply 4 by itself 3 times:- 4 4 4 = 16 4 = 64
- Therefore, 4 3 = 64.
- (b) (-2) 5
**Solution**: In this case, the base is -2, so it should be multiplied by itself 5 times. Don't stress out about the negative signs. Just go left to right, and multiply two numbers at a time. Start with (-2) (-2) to get 4 and then multiply that result by the next -2, and that result by the next -2 until you're finished.- (-2)(-2)(-2)(-2)(-2) = 4(-2)(-2)(-2) = -8(-2)(-2) = 16(-2) = -32

##### Critical Point

Two exponents have special names. Anything raised to the second power is said to be

*squared*(5 2 can be read "5 squared"), and anything to the third power is said to be*cubed*(x 3 can be read "x cubed").##### You've Got Problems

Problem 2: Evaluate the expression: (-3) 4 .

### Exponential Rules

Once you write something in exponential form, there are very specific rules you must follow to simplify expressions. Here are the five most important rules, each with a brief explanation:

**Rule 1**:**x****a****x**=**b****x****a**+**b**. If exponential expressions with the same base are multiplied, the result is the common base raised to the*sum*of the powers.

**Rule 2**:**x**a_{x b}=**x**. If you are dividing exponential expressions with the same base, the result is the common base raised to the*a-b**difference*of the two powers.

*z*7_{z 4}=*z*7 - 4 = z 3 (-5) 10_{(-5) 9}= (-5) 1 = -5

##### Critical Point

Any number raised to the 1 power equals the original number (

*x*1 =*x*) so, if there's no power written, it's understood to be 1 (7 = 7 1 ). In addition, anything (except 0) raised to the 0 power equals 1 (x 0 = 1, 12 0 = 1). The expression 0 0 works a little differently, but you don't deal with that until calculus.**Rule 3**: (**x****a**) b =**x****a b**. If an exponential expression is itself raised to a power, multiply the exponents together. This is different from Rule 1, because here there is one base raised to two powers, and in Rule 1, there were two bases raised to two powers.

- (3 5 ) 6 = 3 5 6 = 3 30 (
*k*2 ) 0 =*k*2 0 =*k*0 = 1

**Rule 4**: ()*xy*=**a***x***a***y***a****and**( x_{y}) a =a**x**_{y a}. If a product (multiplication problem) or quotient (division problem) or any such combination is raised to a power, then so is every individual piece within.

- (5
*y*) 2 = 5 2*y*2 = 25*y*2 (x 2 y 3 ) 4 = (*x*2 ) 4 (*y*3 ) 4 =*x*8*y*12

**Rule 5**:**x****-a**=**1**_{x -a}**and****1**_{x -a}=**x**. If something is raised to a negative power, move it to the other part of the fraction (if it's in the numerator, send it to the denominator and vice versa) and change the exponent to its opposite. If the expression contains other positive exponents, leave them alone.**a**

*Most teachers consider answers containing negative exponents unsimplified*, so make sure to eliminate negative exponents from your final answer. Also, note that raising something to the -1 power is equivalent to taking its reciprocal.

Most of the time, you'll have to apply multiple rules during the same problem, in your attempts to simplify.

**Example 3**: Simplify the expression (*x*2*y*-3 ) 2_{(xy 2 ) 4}.**Solution**: Start by applying Rules 3 and 4 to the numerator and denominator.Now apply Rule 2, since you have matching bases in the numerator and denominator.

##### You've Got Problems

Problem 3: Simplify the expression (

*x*3 y) 5 (*x*-2 y 2 ) 3 .Finish by applying Rule 5.

Excerpted from The Complete Idiot's Guide to Algebra 2004 by W. Michael Kelley. All rights reserved including the right of reproduction in whole or in part in any form. Used by arrangement with

**Alpha Books**, a member of Penguin Group (USA) Inc.

## MathHelp.com

#### Simplify the following expression:

The exponent rules tell me to subtract the exponents. But let's suppose that I've forgotten the rules again. The " 6 8 " means I have eight copies of 6 on top the " 6 5 " means I have five copies of 6 underneath.

How many extra 6 's do I have, and where are they? I have three extra 6 's, and they're on top. Then:

Unless the instructions also tell you to "evaluate", you're probably expected to leave numerical exponent problems like this in exponent form. If you're not sure, though, feel free to add " = 216 ", just to be on the safe side.

#### Simplify the following expression:

How many extra copies of

*t*do I have, and where are they? I have two extra copies, on top:Once you become comfortable with the "how many extras do I have, and where are they?" reasoning, you'll find yourself not needing to write things out and cancel off the duplicate factors. The answers will start feeling fairly obvious to you.

#### Simplify the following expression:

This question is a bit different, because the larger exponent is on the term in the denominator. But the basic reasoning is the same.

How many extra copies of 5 do I have, and where are they? I have six extra copies, and they're underneath:

Note: If you apply the subtraction rule, you'll end up with 5 3&ndash9 = 5 &ndash6 , which is mathematically correct, but is almost certainly not the answer they're looking for.

Whether or not you've been taught about negative exponents, when they say "simplify", they mean "simplify the expression so it doesn't have any negative or zero powers". Some students will try to get around this minus-sign problem by arbitrarily switching the sign to magically get " 5 6 " on top (rather than below a " 1 "), but this is incorrect.

Let's move on to expressions that are a bit more complex.

#### Simplify the following expression:

I mustn't forget that the " 5 " and the " 3 " are just numbers. Since 3 doesn't go evenly into 5 , I can't cancel the numbers.

And I mustn't try to subtract the numbers, because the 5 and the 3 in the fraction " " are not at all the same as the 5 and the 3 in rational expression " ". The numerical portion stays as it is.

For the variables, I have two extra copies of

*x*on top, so the answer is:Either of the purple highlighted answers should be acceptable: the only difference is in the formatting they mean the same thing.

#### Simplify (&ndash46x 2 y 3 z) 0

This is simple enough: anything to the zero power is just 1 .

#### Simplify &ndash(46x 2 y 3 z) 0

The parenthetical portion still simplifies to 1 , but this time the "minus" is out in front of the parentheses that is, it's out from under the power, so the exponent doesn't touch it. So the answer in this case is:

#### Simplify the following expression:

I can cancel off the common factor of 5 in the numerical part of the fraction:

Now I need to look at each of the variables. How many extra of each do I have, and where are they? I have two extra

*a*'s on top. I have one extra*b*underneath. And I have the same number of*c*'s top and bottom, so they'll cancel off entirely. This gives me:

## A REVIEW OF LOGARITHMS

Before you choose a topic from the list, let's review what you know about the word "equivalent." You will come across this word in all your math courses. According to Webster, the word "equivalent" means to have equal value, equal amount, equal measure, equal area, equal length, etc. You get the picture.

For example, 1 yard is equivalent to 3 feet because both represent the same measure. You will see the notation 1 yard = 3 feet. Obviously, the left side of the = sign is not identical to the right side. The numbers and the words are different. However, the left side is equivalent to the right side because both sides denote the same length. In this case, the = sign is used to denote equivalence.

Suppose you wish to purchase an item for , you could pay for it with a dollar bill and 2 dimes, you could pay for it with 12 dimes, you could pay it for 4 quarters and 4 nickels, etc. If you put each method of payment in a separate pile, you will note that they do not look the same. They are in fact not identical. However, they are equivalent because all the methods of payment total .The number 1 has many equivalent faces:

to name a few. The number 0 has many equivalent faces:

Exponents represent shortcuts in multiplication.

Example 1: The expression can be written .

Example 2: The expression means .

Example 3: The expression can be written .

Example 4: The expression (-7)(-7)(-7)(-7) can be written .

You would much rather write the expression than write out 8 one thousand times. You can also use exponents with variables.

Example 5: The expression ( a + b )( a + b )( a + b )( a + b )( a + b ) is more conveniently written as .

If you want to review the rules for exponents, click on Exponential Rules.

## Exploring Exponents

Since this is the beginning of a new unit, I want to find out what students know about the vocabulary that goes along with expressions. I will be having them complete a vocabulary rating inventory sheet. This sheet has the important vocabulary for this unit. Students will look at the vocabulary words and make a decision on how much they know about each word. As you move through the unit, have students re-visit the words to see if they have moved along on the inventory sheet. Students should place this in front of the notes for this unit.

To help students out with this rating scale, read the words aloud as the students complete the inventory for each word. This inventory supports

**mathematical practices 1 and 6.**Students need to explain and ask “does this make sense”. Also, using vocabulary accurately and precisely in their communications.

## 6.2: What Happens with Zero and Negative Exponents? (15 minutes)

### Activity

Extend the definition of non-positive exponents to other bases. The work of this activity is parallel to the work with the powers of 10 table in the previous lesson. Notice students who transfer their understanding of powers of 10, especially non-positive exponents, to exponential expressions with bases other than 10.

### Launch

Arrange students in groups of 2. Tell students to work on their own and then share their reasoning with their partner after each has had a chance to complete the problem. Encourage students to look for similarities to work they have already done with base 10. Give students 10 minutes of work time, followed by a whole class discussion.

Complete the table to show what it means to have an exponent of zero or a negative exponent.

Expand Image

**Description:**<p>A table with two rows. First row, value, 16, 4 blanks, fraction 1 over 2, three blanks. Second row, exponent form, 2 to the fourth power, 8 blanks. Above the top row, arrows point column to column right to left, labeled time 2. Below the bottom row, arrows point column to column, left to right, labeled times question mark.</p>- As you move toward the left, each number is being multiplied by 2. What is the multiplier as you move toward the right?
- Use the patterns you found in the table to write (2^< ext -6>) as a fraction.
- Write (frac<1><32>) as a power of 2 with a single exponent.
- What is the value of (2^0) ?
- From the work you have done with negative exponents, how would you write (5^< ext -3>) as a fraction?
- How would you write (3^< ext -4>) as a fraction?

### Student Response

Teachers with a valid work email address can click here to register or sign in for free access to Student Response.

#### Are you ready for more?

Find an expression equivalent to (left(frac<2><3> ight)^< ext-3>) but with positive exponents.

Find an expression equivalent to (left(frac<4><5> ight)^< ext-8>) but with positive exponents.

What patterns do you notice when you start with a fraction raised to a negative exponent and rewrite it using a single positive exponent? Show or explain your reasoning.

### Student Response

Teachers with a valid work email address can click here to register or sign in for free access to Extension Student Response.

### Activity Synthesis

The important idea for this activity is that the same logic that we used to analyze powers of 10 applies to other bases as well. Students will work mostly with bases that are whole numbers and positive fractions.

Here are some sample questions for whole-class discussion:

- “How does working with powers of 2 compare to working with powers of 10?” (The properties apply in the same way, but computing the value of the exponents is more difficult.)
- “What happens when you use exponent rules when one of the exponents is 0?” (The rules apply in the same way. (2^0 = 1) in the same way that (10^0 = 1) .)
- “How does (10^< ext- 3>) compare to (2^< ext- 3>) ?” (Both values are positive and less than 1. Since (10^< ext- 3>) has greater numbers in the denominator, it is less than (2^< ext- 3>) .)

Introduce new (or update current) visual displays to illustrate exponent rules for positive, rational bases. Display them for all to see throughout the unit. An example would be to update the rule (10^n oldcdot 10^m = 10^

) with (x^n oldcdot x^m = x^ ) with a guiding example that uses a base other than 10.

## 6.3.4: Equivalent Exponential Expressions - Mathematics

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