Q. 8.50 - Intermediate Algebra

Question

Simplify:

$\left(a\right) \frac{\sqrt{128m^9}}{\sqrt{2m}} \left(b\right) \frac{\sqrt[3]{-192}}{\sqrt[3]{3}} \left(c\right) \frac{\sqrt[4]{324n^7}}{\sqrt[4]{2n^3}}.$

Step-by-Step Solution

Answer

Part a. After simplifying we get, $8m^4.$

Part b. After simplifying we get, $-4.$

Part c. After simplifying we get, $3n \sqrt[4]{2}.$

Part a Step 1. Use the Quotient property to write as one radical

Given: $\frac{\sqrt{128m^9}}{\sqrt{2m}}$

By using the property we can write as $\sqrt{\frac{128m^9}{2m}}.$

Part a Step 2: Simplify the fraction under the radical

By simplifying we get,

$$\begin{gathered} \sqrt{\frac{128m^9}{2m}} \\ =\sqrt{64m^{9-1}} \\ =\sqrt{{\left(8m^4\right)}^2} \\ =8m^4 \end{gathered}$$

Part b Step 1. Use the Quotient property to write as one radical

Given: $\frac{\sqrt[3]{-192}}{\sqrt[3]{3}}$

By using the property we can write as $\sqrt[3]{\frac{-192}{3}}.$

Part b Step 2: Simplify the fraction under the radical

By simplifying we get,

$$\begin{gathered} \sqrt[3]{\frac{-192}{3}} \\ =\sqrt[3]{-64} \\ =\sqrt[3]{-{\left(4\right)}^3} \\ =-4 \end{gathered}$$

Part c Step 1. Use the Quotient property to write as one radical

Given: $\frac{\sqrt[4]{324n^7}}{\sqrt[4]{2n^3}}$

By using the property we can write as $\sqrt[4]{\frac{324n^7}{2n^3}}.$

Part c Step 2: Simplify the fraction under the radical

By simplifying we get,

$$\begin{gathered} \sqrt[4]{\frac{324n^7}{2n^3}} \\ =\sqrt[4]{162n^{7-3}} \\ =\sqrt[4]{162n^4} \\ =\sqrt[4]{{\left(3n\right)}^4}·\sqrt[4]{2} \\ =3n \sqrt[4]{2} \end{gathered}$$