Q 137E - Intermediate Algebra

Question

In the following exercises, simplify.

Part (a): ${\left(-36\right)}^{\frac{1}{2}}$

Part (b): $-{36}^{\frac{1}{2}}$

Part (c): ${\left(36\right)}^{\frac{-1}{2}}$

Step-by-Step Solution

Answer

Part (a): On simplifying, we know the solution is not real.

Part (b): On simplifying, we get $-6$.

Part (c): On simplifying, we get $\frac{1}{6}$.

Part (a) Step 1. Rewrite as a root.

Consider the given question,

${\left(-36\right)}^{\frac{1}{2}}$

Rewrite as a square root,

$=\sqrt[2]{{\left(-36\right)}^2}$

The solution is not real.

Part (b) Step 1. Rewrite as a root.

Consider the given question,

$-{36}^{\frac{1}{2}}$

The exponent only applies to the $36$.

Rewrite as a square root,

$=-\sqrt[2]{36}$

On simplifying the expression,

$$\begin{gathered} =-\sqrt[2]{6^2} \\ =-6 \end{gathered}$$

Part (c) Step 1. Rewrite as a root.

Consider the given question,

${\left(36\right)}^{\frac{-1}{2}}$

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

$=\frac{1}{{\left(36\right)}^{{\displaystyle \frac{1}{2}}}}$

Rewrite as a square root,

$=\frac{1}{\sqrt[2]{36}}$

On simplifying the expression,

$$\begin{gathered} =\frac{1}{\sqrt[2]{6^2}} \\ =\frac{1}{6} \end{gathered}$$