Chapter 4.3, Problem 6E

Solution

To calculate: The value of ${(2e^{-2})}^{-4}$.

The solution of the given problem is $\frac{1}{16}{e}^8$

Given information: Compute the given expression ${(2e^{-2})}^{-4}$.

Calculation: Step 1: Check which property of exponents goes with the given question.

Here, property of power raised to a power meets which states that if a is a base then the power raised to the power of base a gives the product of powers raised to the base a. it means ${\left(a^m\right)}^n=a^{mn}$

Solve this problem in steps.

Step 2: Write the problem in two parts.

First part $2^{-4}$ and $e^{-2\left(-4\right)}$

Step 2: Check what are the coefficients, exponents and base.

An exponent is a number or letter written above and to the right of a mathematical expression called the base.

Coefficient is a number multiplied by a variable.

A base is a number which gets multiplied when using an exponent.

Here, base is $e^{}$.

The coefficient is $2^{-4}$.

  $\begin{array}{l}{2}^{-4}=\frac{1}{2^4}\\ =\frac{1}{16}\end{array}$

And the other part is simplified as:

  $\begin{array}{l}{e}^{-2\left(-4\right)}=e^8\\ \end{array}$

The final answer for the given expression is $\frac{1}{16}{e}^8$.