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5.5: Laws of Exponents

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In Chapter 1, section 1, we first introduced the definition of an exponent. For convenience, we repeat that definition.

In the exponential expression an, the number a is called the base, while the number n is called the exponent.

Exponents

Let a be any real number and let n be any whole number. If n0, then:

an=aaaan times 

That is, to calculate an, write a as a factor n times. In the case where a0, but n=0, then we define:

a0=1

For example, raising a number to the fifth power requires that we repeat the number as a factor five times (see Figure 5.5.1).

(2)5=(2)(2)(2)(2)(2)=32

fig 5.5.1.png
Figure 5.5.1: Evaluating (2)5 and (1/2)4. Recall: Use the MATH key to locate the ►Frac command.

Raising a number to the fourth power requires that we repeat that number as a factor four times (see Figure 5.5.1).

(12)4=(12)(12)(12)(12)=116

As a final example, note that 100=1, but 00 is undefined (see Figure 5.5.2).

fig 5.5.2.png
Figure 5.5.2: Evaluating 100 and 00 on the graphing calculator.

Note

For those who may be wondering why a0=1, provided a0, here is a nice argument. First, note that a1=a, so:

aa0=a1a0

On the right, repeat the base and add the exponents.

aa0=a1

Or equivalently:

aa0=a

Now, divide both sides by a, which is permissible if a0.

aa0a=aa

Simplify both sides:

a0=1

Multiplying With Like Bases

In the expression an, the number a is called the base and the number n is called the exponent. Frequently, we’ll be required to multiply two exponential expressions with like bases, such as x3x4. Recall that the exponent tells us how many times to write each base as a factor, so we can write:

x3x4=(xxx)(xxxx)=xxxxxxx=x7

Note that we are simply counting the number of times that x occurs as a factor. First, we have three xs, then four xs, for a total of seven xs. However a little thought tells us that it is much quicker to simply add the exponents to reveal the total number of times x occurs as a factor.

x3x4=x3+4=x7

The preceding discussion is an example of the following general law of exponents.

Multiplying With Like Bases

To multiply two exponential expressions with like bases, repeat the base and add the exponents.

aman=am+n

Example 5.5.1

Simplify each of the following expressions:

  1. y4y8
  2. 2325
  3. (x+y)2(x+y)7

Solution

In each example we have like bases. Thus, the approach will be the same for each example: repeat the base and add the exponents.

  1. y4y8=y4+8=y12
  2. 2325=23+5=28
  3. (x+y)2(x+y)7=(x+y)2+7=(x+y)9

With a little practice, each of the examples can be simplified mentally. Repeat the base and add the exponents in your head: y4y8=y12,2325=28 and (x+y)2(x+y)7=(x+y)9.

Exercise 5.5.1

3432

Answer

36

Example 5.5.2

Simplify: (a6b4)(a3b2)

Solution

We’ll use the commutative and associative properties to change the order of operation, then repeat the common bases and add the exponents.

(a6b4)(a3b2)=a6b4a3b2 The associative property allows us to regroup in the order we prefer. =a6a3b4b2 The commutative property allows us to change the order of multiplication. =a9b6 Repeat the common bases and add the exponents. 

With practice, we realize that if all of the operators are multiplication, then we can multiply in the order we prefer, repeating the common bases and adding the exponents mentally: (a6b4)(a3b2)=a9b6.

Exercise 5.5.2

(x2y6)(x4y3)

Answer

x6y9

Example 5.5.3

Simplify: xn+3x32n

Solution

Again, we repeat the base and add the exponents.

xn+3x32n=x(n+3)+(32n) Repeat the base, add the exponents. =x6n Simplify. Combine like terms. 

Exercise 5.5.3

x5nx4n+2

Answer

x3n+7

Dividing With Like Bases

Like multiplication, we will also be frequently asked to divide exponential expressions with like bases, such as x7/x4. Again, the key is to remember that the exponent tells us how many times to write the base as a factor, so we can write:

x7x4=xxxxxxxxxxx=xxxxxxxxxxx=x3

Note how we cancel four xs in the numerator for four xs in the denominator. However, in a sense we are “subtracting four xs” from the numerator, so a faster way to proceed is to repeat the base and subtract the exponents, as follows:

x7x4=x74=x3

The preceding discussion is an example of the second general law of exponents.

How to Divide with Like Bases

To divide two exponential expressions with like bases, repeat the base and subtract the exponents. Given a0,

aman=amn

Note that the subtraction of the exponents follows the rule “top minus bottom.”

Note

Here is another nice argument why a0=1, provided a0. Start with:

a1a1=1

Repeat the base and subtract the exponents.

a11=1

Simplify.

a0=1

Example 5.5.4

Simplify each of the following expressions:

  1. x12x3
  2. 5757
  3. (2x+1)8(2x+1)3

Solution

In each example we have like bases. Thus, the approach will be the same for each example: repeat the base and subtract the exponents.

  1. x12x3=x123=x9
  2. 5757=577=50=1
  3. (2x+1)8(2x+1)3=(2x+1)83=(2x+1)5

With a little practice, each of the examples can be simplified mentally. Repeat the base and subtract the exponents in your head: x12/x3=x9,57/54=53 and (2x+1)8/(2x+1)3=(2x+1)5.

Exercise 5.5.4

4543

Answer

42

Example 5.5.5

Simplify: 12x5y74x3y2

Solution

We first express the fraction as a product of three fractions, the latter two with a common base. In the first line of the following solution, note that if you multiply numerators and denominators of the three separate fractions, the product equals the original fraction on the left.

12x5y74x3y2=124x5x3y7y2 Break into a product of three fractions. =3x53y72 Simplify: 12/4=3. Then repeat the common =3x2y5 Simplify. 

Exercise 5.5.5

Simplify: 15a6b93ab5

Answer

5a5b4

Example 5.5.6

Simplify: x5n4x32n

Solution

Again, we repeat the base and subtract the exponents.

x5n4x32n=x(5n4)(32n) Repeat the base, subtract exponents. =x5n43+2n Distribute the minus sign. =x7n7 Simplify. Combine like terms. 

Exercise 5.5.6

Simplify: x3n6xn+2

Answer

x2n8

Raising a Power to a Power

Suppose we have an exponential expression raised to a second power, such as (x2)3. The second exponent tells us to write x2 as a factor three times:

(x2)3=x2x2x2 Write x2 as a factor three times. =x6 Repeat the base, add the exponents. 

Note how we added 2+2+2 to get 6. However, a much faster way to add “three twos” is to multiply: 32=6. Thus, when raising a “power to a second power,” repeat the base and multiply the exponents, as follows:

(x2)3=x23=x6

The preceding discussion gives rise to the following third law of exponents.

Raising a Power to a Power

When raising a power to a power, repeat the base and multiply the exponents. In symbols:
(am)n=amn
Note that juxtaposing two variables, as in mn, means “m times n.”

Example 5.5.7

Simplify each of the following expressions:

  1. (z3)5
  2. (73)0
  3. [(xy)3]6

Solution

In each example we are raising a power to a power. Hence, in each case, we repeat the base and multiply the exponents.

  1. (z3)5=z3.5=z15
  2. (73)0=73.0=70=1
  3. [(xy)3]6=(xy)36=(xy)18

With a little practice, each of the examples can be simplified mentally. Repeat the base and multiply the exponents in your head: (z3)5=z15,(73)4=712 and [(xy)3]6=(xy)18.

Exercise 5.5.7

Simplify: (23)4

Answer

212

Example 5.5.8

Simplify: (x2n3)4

Solution

Again, we repeat the base and multiply the exponents.

(x2n3)4=x4(2n3) Repeat the base, multiply exponents. =x8n12 Distribute the 4.

Exercise 5.5.8

Simplify: (a2n)3

Answer

a63n

Raising a Product to a Power

We frequently have need to raise a product to a power, such as (xy)3. Again, remember the exponent is telling us to write xy as a factor three times, so:

(xy)3=(xy)(xy)(xy)Write xy as a factor three times.=xyxyxyThe associative property allows us to group as we please.=xxxyyyThe commutative property allows us to change the order as we please.=x3y3Invoke the exponent definition: xxx=x3 and yyy=y3

However, it is much simpler to note that when you raise a product to a power, you raise each factor to that power. In symbols: (xy)3=x3y3
The preceding discussion leads us to a fourth law of exponents.

Raising a Product to a Power

To raise a product to a power, raise each factor to that power. In symbols:

(ab)n=anbn

Example 5.5.9

Simplify each of the following expressions:

  1. (yz)5
  2. (2x)3
  3. (3y)2

Solution

In each example we are raising a product to a power. Hence, in each case, we raise each factor to that power.

  1. (yz)5=y5z5
  2. (2x)3=(2)3x3=8x3
  3. (3y)2=(3)2y2=9y2

With a little practice, each of the examples can be simplified mentally. Raise each factor to the indicated power in your head: (yz)5=y5z5,(2x)3=8x3 and (3y)2=9y2

Exercise 5.5.9

Simplify: (2b)4

Answer

16b4

When raising a product of three factors to a power, it is easy to show that we should raise each factor to the indicated power. For example, (abc)3=a3b3c3. In general, this is true regardless of the number of factors. When raising a product to a power, raise each of the factors to the indicated power.

Example 5.5.10

Simplify: (2a3b2)3

Solution

Raise each factor to the third power, then simplify.

(2a3b2)3=(2)3(a3)3(b2)3 Raise each factor to the third power. =8a9b6Simplify: (2)3=8. In the remaining factors, raising a power to a power requires that we multiply the exponents. 

Exercise 5.5.10

Simplify: (3xy4)5

Answer

243x5y20

Example 5.5.11

Simplify: (2x2y)2(3x3y)

Solution

In the first grouped product, raise each factor to the second power.

(2x2y)2(3x3y)=((2)2(x2)2y2)(3x3y) Raise each factor in the first grouped product to the second power.=(4x4y2)(3x3y) Simplify: (2)2=4 and (x2)2=x4

The associative and commutative property allows us to multiply all six factors in the order that we please. Hence, we’ll multiply 4 and 3, then x4 and x3, and \(y^2 and y, in that order. In this case, we repeat the base and add the exponents.

=12x7y3 Simplify: (4)(3)=12. Also, x4x3=x7 and y2y=y3

Exercise 5.5.11

Simplify: (a3b2)3(2a2b4)2

Answer

4a13b14

Raising a Quotient to a Power

Raising a quotient to a power is similar to raising a product to a power. For example, raising (x/y)3 requires that we write x/y as a factor three times.

(xy)3=xyxyxy=xxxyyy=x3y3

However, it is much simpler to realize that when you raise a quotient to a power, you raise both numerator and denominator to that power. In symbols:

(xy)3=x3y3

This leads to the fifth and final law of exponents.

Raising a Quotient to a Power

To raise a quotient to a power, raise both numerator and denominator to that power. Given b0,

(ab)n=anbn

Example 5.5.12

Simplify each of the following expressions:

  1. (23)2
  2. (x3)3
  3. (2y)4

Solution

In each example we are raising a quotient to a power. Hence, in each case, we raise both numerator and denominator to that power.

  1. (23)2=2232=49
  2. (x3)3=x333=x327
  3. (2y)4=24y4=16y4

Note that in example (c), raising a negative base to an even power produces a positive result. With a little practice, each of the examples can be simplified mentally. Raise numerator and denominator to the indicated power in your head: (2/3)2=4/9,(x/3)3=x3/27, and (2/y)4=16/y4

Exercise 5.5.12

Simplify: (54)3

Answer

12564

Example 5.5.13

Simplify: (2x5y3)2

Solution

Raise both numerator and denominator to the second power, then simplify:

(2x5y3)2=(2x5)2(y3)2Raise numerator and denominator to the second power.

In the numerator, we need to raise each factor of the product to the second power. Then we need to remind ourselves that when we raise a power to a power, we multiply the exponents.

=22(x5)2(y3)2Raise each factor in the numerator and denominator to the second power.=4x10y6 Simplify: 22=4,(x5)2=x10, and (y3)2=y6

Exercise 5.5.13

Simplify: (a43b2)3

Answer

a1227b6


This page titled 5.5: Laws of Exponents is shared under a CC BY-NC-ND 3.0 license and was authored, remixed, and/or curated by David Arnold via source content that was edited to the style and standards of the LibreTexts platform.

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