Question

In the following exercises, simplify.

a.$\sqrt{48b^5}-\sqrt{75b^5}$

b.$8\sqrt[3]{64q^6}-3\sqrt[3]{125q^6}$

Short answer

On simplification, we get$-b^2\sqrt{3b},17q^2$respectively

Step-by-step solution

Part (a) Step 1: Given information

Given the expression$\sqrt{48b^5}-\sqrt{75b^5}$

Part (a) Step 2: Simplifying the radical and combine like terms

Simplifying radical and combining we get

$$\begin{gathered} \sqrt{48b^5}-\sqrt{75b^5}=\sqrt{16b^4·3b}-\sqrt{25b^4·3b} \\ \sqrt{48b^5}-\sqrt{75b^5}=\sqrt{16b^4}·\sqrt{3b}-\sqrt{25b^4}·\sqrt{3b} \\ \sqrt{48b^5}-\sqrt{75b^5}=4b^2\sqrt{3b}-5b^2\sqrt{3b} \\ \sqrt{48b^5}-\sqrt{75b^5}=-b^2\sqrt{3b} \end{gathered}$$

Part (b) Step 1: Given information

Given expression $8\sqrt[3]{64q^6}-3\sqrt[3]{125q^6}$

Part (b) Step 2: Simplifying the radical and combine like terms 

Simplifying radical and combining we get

$$\begin{gathered} 8\sqrt[3]{64q^6}-3\sqrt[3]{125q^6}=8\sqrt[3]{64q^6}-3\sqrt[3]{125q^6} \\ 8\sqrt[3]{64q^6}-3\sqrt[3]{125q^6}=8·\sqrt[3]{{\left(4q^2\right)}^3}-3·\sqrt[3]{{\left(5q^2\right)}^3} \\ 8\sqrt[3]{64q^6}-3\sqrt[3]{125q^6}=8·4q^2-3·5q^2 \\ 8\sqrt[3]{64q^6}-3\sqrt[3]{125q^6}=32q^2-15q^2 \\ 8\sqrt[3]{64q^6}-3\sqrt[3]{125q^6}=17q^2 \end{gathered}$$