Q. 179 - Intermediate Algebra

Question

In the following exercises, simplify.

a.$\sqrt{80c^7}-\sqrt{20c^7}$

b.$2\sqrt[4]{162r^{10}}+4\sqrt[4]{32r^{10}}$

Step-by-Step Solution

Answer

On simplification, we get $2c^3\sqrt{5c},14r^2\sqrt[4]{2r^2}$ respectively

Part (a) Step 1: Given information

Given expression $\sqrt{80c^7}-\sqrt{20c^7}$

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

Simplifying radical and combining we get

$$\begin{gathered} \sqrt{80c^7}-\sqrt{20c^7}=\sqrt{16c^6·5c}-\sqrt{4c^6·5c} \\ \sqrt{80c^7}-\sqrt{20c^7}=\sqrt{16c^6}·\sqrt{5c}-\sqrt{4c^6}·\sqrt{5c} \\ \sqrt{80c^7}-\sqrt{20c^7}=\sqrt{{\left(4c^3\right)}^2}·\sqrt{5c}-\sqrt{{\left(2c^3\right)}^2}·\sqrt{5c} \\ \sqrt{80c^7}-\sqrt{20c^7}=4c^3\sqrt{5c}-2c^3\sqrt{5c} \\ \sqrt{80c^7}-\sqrt{20c^7}=2c^3\sqrt{5c} \end{gathered}$$

Part (b) Step 1: Given information

Given expression $2\sqrt[4]{162r^{10}}+4\sqrt[4]{32r^{10}}$

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

Simplifying radical and combining we get

$$\begin{gathered} 2\sqrt[4]{162r^{10}}+4\sqrt[4]{32r^{10}}=2\sqrt[4]{81r^8·2r^2}+4\sqrt[4]{16r^8·2r^2} \\ 2\sqrt[4]{162r^{10}}+4\sqrt[4]{32r^{10}}=2·\sqrt[4]{81r^8}·\sqrt[4]{2r^2}+4·\sqrt[4]{16r^8}.\sqrt[4]{2r^2} \\ 2\sqrt[4]{162r^{10}}+4\sqrt[4]{32r^{10}}=2·\sqrt[4]{{\left(3r^2\right)}^4}·\sqrt[4]{2r^2}+4·\sqrt[4]{{\left(2r^2\right)}^4}·\sqrt[4]{2r^2} \\ 2\sqrt[4]{162r^{10}}+4\sqrt[4]{32r^{10}}=2·3r^2·\sqrt[4]{2r^2}+4·2r^2·\sqrt[4]{2r^2} \\ 2\sqrt[4]{162r^{10}}+4\sqrt[4]{32r^{10}}=6r^2\sqrt[4]{2r^2}+8r^2\sqrt[4]{2r^2} \\ 2\sqrt[4]{162r^{10}}+4\sqrt[4]{32r^{10}}=14r^2\sqrt[4]{2r^2} \end{gathered}$$