Q. 118 - Intermediate Algebra

Question

Explain why $\sqrt[4]{-64}$ is not a real number but $\sqrt[3]{-64}$ is.

Step-by-Step Solution

Answer

$-64$ is not a perfect fourth power while $-64$ is a perfect third power. So $\sqrt[3]{-64}$ is a real number and $\sqrt[4]{-64}$ is not a real number.

Step 1. Given

To explain why $\sqrt[4]{-64}$ is not a real number while $\sqrt[3]{-64}$ is a real number.

Step 2. Consider - 64 4

Consider the expression $\sqrt[4]{-64}$

This expression is not a perfect fourth power.

So, it is not a real number.

Step 3. Consider - 64 3

Consider the expression $\sqrt[3]{-64}$

$\sqrt[3]{-64}=\sqrt[3]{-{\left(4\right)}^3}$

           $=-4$

So $\sqrt[4]{-64}$ is not a real number while $\sqrt[3]{-64}$  is a real number.