Question

Simplify:

(a)${(-64)}^{-\frac{1}{2}}$

(b) ${-64}^{\frac{1}{2}}$

(c) ${\left(64\right)}^{-\frac{1}{2}}$

Short answer

The simplification of ${(-64)}^{-\frac{1}{2}}$ is no real root

The simplification of $-{64}^{\frac{1}{2}}$is$-8$

The simplification of ${\left(64\right)}^{-\frac{1}{2}}$is $\frac{1}{8}$

Step-by-step solution

Part (a) Step 1. Given

The expression is ${(-64)}^{\frac{1}{2}}$

To simplify the expression.

Part (a) Step 2. Rewrite as the nth root

Rewrite using the property, $a^{-n}=\frac{1}{a^n}$,

${(-64)}^{\frac{-1}{2}}=\frac{1}{{(-64)}^{{\displaystyle \frac{1}{2}}}}$
Rewrite as the square root

It cannot be written as perfect square root.

So it has no real root.

Part (b) Step 1. Given

The expression is $-{64}^{\frac{1}{2}}$

To simplify the expression.

Part (b) Step 2. Rewrite as the nth root

Rewrite as the square root

$-{64}^{\frac{1}{2}}$

The root is only for $64$. So,

$-{64}^{\frac{1}{2}}=-{\left(8^2\right)}^{\frac{1}{2}}$

$=-8$

Part (c) Step 1. Given

The expression is ${\left(64\right)}^{-\frac{1}{2}}$

To simplify the expression.

Part (c) Step 2. Rewrite as the nth root

Rewrite using the property, $a^{-n}=\frac{1}{a^n}$

${\left(64\right)}^{-\frac{1}{2}}=\frac{1}{64\frac{1}{2}}$

Rewrite as the square root

$=\frac{1}{{\left(8^2\right)}^{{\displaystyle \frac{1}{2}}}}$

$=\frac{1}{8}$