Chapter 7.3, Problem 62PPE

To determine

To simplify: The expression ${\left(x^2\right)}^3$ and $x^{2^3}$ . Are these expression are equivalent. Explain.

Answer to Problem 62PPE

The expression ${\left(x^2\right)}^3$ and $x^{2^3}$ are not equivalent because they have the different powers.

Explanation of Solution

Given information:

The given expression ${\left(x^2\right)}^3$ and $x^{2^3}$ .

Calculation:

  $\begin{array}{l}={\left(x^2\right)}^3\\ =x^{2\cdot 3}\end{array}$

(Multiply exponents when raising a power to a power)

  $=x^6$

(Simplify)

Again,

  $\begin{array}{l}=x^{2^3}\\ =x^8\end{array}$

(Simplify)

Therefore, the expression ${\left(x^2\right)}^3$ and $x^{2^3}$ are not equivalent because they have the different powers.