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Powers of a Monomial

What happens when a monomial is raised to a power? A good way to introduce this concept is to work again with powers of 2. First look at the squares of various powers of 2.

2 n ( 2 n )2
2 1 = 2 4 = 2 2
2 2 = 4 16 = 2 4
2 3 = 8 64 = 2 6
2 4 = 16 256 = 2 8
2 5 = 32 1024 = 2 10
2 6 = 64 4096 = 2 12

Notice that when a power of 2 is squared, the exponent in the result is doubled.

Example 1

Simplify ( x 4 ) 3.

Solution

To simplify the expression, write out the powers as products.

In this case, the exponent 4 is tripled. The exponent in the final answer is 12, since the monomial is raised to the third power. This works in general.

Power of a Power

When a power of a number or variable is raised to another power, the result is that same number or variable whose exponent is the product of the exponents.

( x m) n = x mn

 

Example 2

Simplify ( x 8 ) 7 .

Solution

( x 8 ) 7 = x 8 · 7 = x 56

 

Example 3

Simplify (15 3 ) 6 .

Solution

(15 3 ) 6 = 15 3 · 6 = 15 18

It is a good idea that you work out explicitly several examples by expanding the powers into products, so that reasoning behind the key idea is reinforced.

To understand this idea in the general case, take a look at the following diagram.

There are n groups of factors, each of which is itself a group of m factors, so that there are mn factors all together. So, the result is x mn .

You have now learned how to find powers of powers. Since monomials are generally products of powers, it will be easy now to learn how to find powers of products. This will be helpful in finding powers of monomials.

How do you think te expression ( a · b ) 2 should be simplified?

The following solution provides a good explanation.

( a · b ) 2 = ( a · b ) · ( a · b )
  = a · b · a · b
  = a · a · b · b
  = a 2 · b 2

This idea is shown below using numbers.

6 3 = 216

6 3 = (2 · 3) 3
  = 2 3 · 3 3
  = 8 · 27
  = 216

 

Power of a Product

A product raised to a power is the product of the factors raised to the given power.

( a · b) m = a m · b m

A good way to show this is to use the following diagram.

The middle expression, there are exactly m factors each of a and b. Now let's combine these two ideas into one.

 

Power of a Monomial

For any whole numbers m, n, and p, ( a m · b n) p = a mp b np.

Now you can evaluate powers and products of any monomials.

 
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