Q61. - Algebra 2

Question

CRITICAL THINKING For Exercises 61 and 62, use the following proof of the Power of a Power Property.

What definition or property allows you to make each step of the proof?

Step-by-Step Solution

Answer

The definition of exponent allows us to make each step of the proof.

Step-1 – Given

The given property is:

Step-2 – To determine

We have to find: What definition or property allows you to make each step of the proof?

Step-3 – Calculation

The definition of exponent is: $x^p=\underset{}{\underset{p}{\underbrace{x\cdot x\cdot \mathrm{......}\cdot x}}}$ 

Using this property, we can write$a^m=\underset{}{\underset{m}{\underbrace{a\cdot a\cdot \mathrm{......}\cdot a}}}$ and $a^n=\underset{}{\underset{n}{\underbrace{a\cdot a\cdot \mathrm{......}\cdot a}}}$.

Then using the associative property of multiplication, we can say:

$\underset{m}{\underbrace{a\cdot a\cdot \mathrm{......}\cdot a}}\cdot \underset{}{\underset{n}{\underbrace{a\cdot a\cdot \mathrm{......}\cdot a}}}=\underset{}{\underset{m+n}{\underbrace{a\cdot a\cdot \mathrm{......}\cdot a}}}$.

Then again using the definition of exponent we get:$\underset{}{\underset{m+n}{\underbrace{a\cdot a\cdot \mathrm{......}\cdot a}}}=a^{m+n}$ .