Question

Simplify:

(a)${(-256)}^{\frac{1}{4}}$

(b)${-256}^{\frac{1}{4}}$

(c) ${\left(256\right)}^{-\frac{1}{4}}$

Short answer

The simplification of ${(-256)}^{\frac{1}{4}}$ has no real root

The simplification of $-{256}^{\frac{1}{4}}$ is $-4$

The simplification of ${\left(256\right)}^{\frac{1}{4}}$is $\frac{1}{4}$

Step-by-step solution

Part (a) Step 1. Given

The expression is${(-256)}^{\frac{1}{4}}$

To simplify the expression.

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

Rewrite as the fourth root.

${(-256)}^{\frac{1}{4}}=\sqrt[4]{-256}$

$=\sqrt[4]{{(-4)}^4}$

It has no real root.

Part (b) Step 1. Given

The expression is$-{256}^{\frac{1}{4}}$

To simplify the expression.

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

Rewrite as the fourth root

$-{256}^{\frac{1}{4}}=-\sqrt[4]{256}$

The root is only for $256$. So,

$=-\sqrt[4]{4^4}$

$=-4$

Part (c) Step 1. Given

The expression is${\left(256\right)}^{-\frac{1}{4}}$

To simplify the expression.

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

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

${\left(256\right)}^{-\frac{1}{4}}=\frac{1}{\left(256\right)}$¹⁴

Rewrite as the fourth root

$=\frac{1}{\sqrt[4]{4^4}}$

$=\frac{1}{4}$