Chapter 6.1, Problem 1E

To determine

To explain:Which definitions or properties were used to simplify the expression ${\left(4^8\cdot 4^{-4}\right)}^{-2}$ .

Answer to Problem 1E

To simplify the given numerical expression, we use Product of Powers Property, Power of a Power Property, and Negative Exponent Property.

Explanation of Solution

Giveninformation:

The numerical expression is ${\left(4^8\cdot 4^{-4}\right)}^{-2}$ .

To simplify the expression that is inside the parentheses, we use the product of powers property. That is,

  $4^8\cdot 4^{-4}=4^{8+(-4)}=4^4$ .

Then, we have ${\left(4^4\right)}^{-2}$ .

To simplify this further, we will use the power of a power property. It results to,

  ${\left(4^4\right)}^{-2}=4^{4\cdot (-2)}=4^{-8}$

Now, to simplify this further, we will use the negative exponent property. It results to

  $4^{-8}=\frac{1}{4^8}=\frac{1}{65536}$ .

Conclusion:

Therefore, to simplify the given numerical expression, we use Product of Powers Property, Power of a Power Property, and Negative Exponent Property.