<< Chapter < Page Chapter >> Page >

Simplify: p 5 · p y 14 · y 29 .

p 6 y 43

Got questions? Get instant answers now!

Simplify: z · z 7 b 15 · b 34 .

z 8 b 49

Got questions? Get instant answers now!

We can extend the Product Property for Exponents to more than two factors.

Simplify: d 4 · d 5 · d 2 .

Solution

d to the fourth power times d to the fifth power times d squared.
Add the exponents, since bases are the same. d to the power of 4 plus 5 plus 2.
Simplify. d to the eleventh power.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Simplify: x 6 · x 4 · x 8 .

x 18

Got questions? Get instant answers now!

Simplify: b 5 · b 9 · b 5 .

b 19

Got questions? Get instant answers now!

Simplify expressions using the power property for exponents

Now let’s look at an exponential expression that contains a power raised to a power. See if you can discover a general property.

x squared, in parentheses, cubed.
What does this mean?
How many factors altogether?
x squared cubed is x squared times x squared times x squared, which is x times x, multiplied by x times x, multiplied by x times x. x times x has two factors. Two plus two plus two is six factors.
So we have x to the sixth power.
Notice that 6 is the product of the exponents, 2 and 3. x squared cubed is x to the power of 2 times 3, or x to the sixth power.

We write:

( x 2 ) 3 x 2 · 3 x 6

We multiplied the exponents. This leads to the Power Property for Exponents.

Power property for exponents

If a is a real number, and m and n are whole numbers, then

( a m ) n = a m · n

To raise a power to a power, multiply the exponents.

An example with numbers helps to verify this property.

( 3 2 ) 3 = ? 3 2 · 3 ( 9 ) 3 = ? 3 6 729 = 729

Simplify: ( y 5 ) 9 ( 4 4 ) 7 .

Solution


y to the fifth power, in parentheses, to the ninth power.
Use the power property, ( a m ) n = a m·n . y to the power of 5 times 9.
Simplify. y to the 45th power.



4 to the fourth power, in parentheses, to the 7th power.
Use the power property. 4 to the power of 4 times 7.
Simplify. 4 to the twenty-eighth power.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Simplify: ( b 7 ) 5 ( 5 4 ) 3 .

b 35 5 12

Got questions? Get instant answers now!

Simplify: ( z 6 ) 9 ( 3 7 ) 7 .

z 54 3 49

Got questions? Get instant answers now!

Simplify expressions using the product to a power property

We will now look at an expression containing a product that is raised to a power. Can you find this pattern?

( 2 x ) 3 What does this mean? 2 x · 2 x · 2 x We group the like factors together. 2 · 2 · 2 · x · x · x How many factors of 2 and of x ? 2 3 · x 3

Notice that each factor was raised to the power and ( 2 x ) 3 is 2 3 · x 3 .

We write: ( 2 x ) 3 2 3 · x 3

The exponent applies to each of the factors! This leads to the Product to a Power Property for Exponents.

Product to a power property for exponents

If a and b are real numbers and m is a whole number, then

( a b ) m = a m b m

To raise a product to a power, raise each factor to that power.

An example with numbers helps to verify this property:

( 2 · 3 ) 2 = ? 2 2 · 3 2 6 2 = ? 4 · 9 36 = 36

Simplify: ( −9 d ) 2 ( 3 m n ) 3 .

Solution


  1. Negative 9 d squared.
    Use Power of a Product Property, ( ab ) m = a m b m . negative 9 squared d squared.
    Simplify. 81 d squared.

  2. 3 m n cubed.
    Use Power of a Product Property, ( ab ) m = a m b m . 3 cubed m cubed n cubed.
    Simplify. 27 m cubed n cubed.
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Simplify: ( −12 y ) 2 ( 2 w x ) 5 .

144 y 2 32 w 5 x 5

Got questions? Get instant answers now!

Simplify: ( 5 w x ) 3 ( −3 y ) 3 .

125 w 3 x 3 −27 y 3

Got questions? Get instant answers now!

Simplify expressions by applying several properties

We now have three properties for multiplying expressions with exponents. Let’s summarize them and then we’ll do some examples that use more than one of the properties.

Properties of exponents

If a and b are real numbers, and m and n are whole numbers, then

Product Property a m · a n = a m + n Power Property ( a m ) n = a m · n Product to a Power ( a b ) m = a m b m

All exponent properties hold true for any real numbers m and n . Right now, we only use whole number exponents.

Simplify: ( y 3 ) 6 ( y 5 ) 4 ( −6 x 4 y 5 ) 2 .

Solution


  1. ( y 3 ) 6 ( y 5 ) 4 Use the Power Property. y 15 · y 20 Add the exponents. y 35


  2. ( −6 x 4 y 5 ) 2 Use the Product to a Power Property. ( −6 ) 2 ( x 4 ) 2 ( y 5 ) 2 Use the Power Property. ( −6 ) 2 ( x 8 ) ( y 10 ) Simplify. 36 x 8 y 10
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Simplify: ( a 4 ) 5 ( a 7 ) 4 ( −2 c 4 d 2 ) 3 .

a 48 −8 c 12 d 6

Got questions? Get instant answers now!

Simplify: ( −3 x 6 y 7 ) 4 ( q 4 ) 5 ( q 3 ) 3 .

81 x 24 y 28 q 29

Got questions? Get instant answers now!

Simplify: ( 5 m ) 2 ( 3 m 3 ) ( 3 x 2 y ) 4 ( 2 x y 2 ) 3 .

Solution


  1. ( 5 m ) 2 ( 3 m 3 ) Raise 5 m to the second power. 5 2 m 2 · 3 m 3 Simplify. 25 m 2 · 3 m 3 Use the Commutative Property. 25 · 3 · m 2 · m 3 Multiply the constants and add the exponents. 75 m 5


  2. ( 3 x 2 y ) 4 ( 2 x y 2 ) 3 Use the Product to a Power Property. ( 3 4 x 8 y 4 ) ( 2 3 x 3 y 6 ) Simplify. ( 81 x 8 y 4 ) ( 8 x 3 y 6 ) Use the Commutative Property. 81 · 8 · x 8 · x 3 · y 4 · y 6 Multiply the constants and add the exponents. 648 x 11 y 10
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Elementary algebra' conversation and receive update notifications?

Ask