Question

Simplify:

$\left(a\right)\frac{\sqrt{98z^5}}{\sqrt{2z}}\left(b\right)\frac{\sqrt[3]{-500}}{\sqrt[3]{2}}\left(c\right)\frac{\sqrt[4]{486m^{11}}}{\sqrt[4]{3m^5}}.$

Short answer

Part a. After simplifying we get,$7z^2.$

Part b. After simplifying we get,$-5\sqrt[3]{2}.$

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

Step-by-step solution

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

Given: $\frac{\sqrt{98z^5}}{\sqrt{2z}}$

By using the property we can write as$\sqrt{\frac{98z^5}{2z}}.$

Part a Step 2: Simplify the fraction under the radical

By simplifying we get,

$$\begin{gathered} \sqrt{\frac{98z^5}{2z}} \\ =\sqrt{49z^{5-1}} \\ =\sqrt{{\left(7z^2\right)}^2} \\ =7z^2 \end{gathered}$$

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

Given:$\frac{\sqrt[3]{-500}}{\sqrt[3]{2}}$

By using the property we can write as$\sqrt[3]{\frac{-500}{2}}.$

Part b Step 2: Simplify the fraction under the radical

By simplifying we get,

$$\begin{gathered} \sqrt[3]{\frac{-500}{2}} \\ =\sqrt[3]{-250} \\ =\sqrt[3]{\left(-5^3\right)·2} \\ =-5\sqrt[3]{2} \end{gathered}$$

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

Given:$\frac{\sqrt[4]{486m^{11}}}{\sqrt[4]{3m^5}}$

By using the property we can write as$\sqrt[4]{\frac{486m^{11}}{3m^5}}.$

Part c Step 2: Simplify the fraction under the radical

By simplifying we get,

$$\begin{gathered} \sqrt[4]{\frac{486m^{11}}{3m^5}} \\ =\sqrt[4]{162m^6} \\ =\sqrt[4]{{\left(3m\right)}^4}·\sqrt[4]{2m^2} \\ =3m\sqrt[4]{2m^2} \end{gathered}$$