Q. 187 - Intermediate Algebra

Question

In the following exercises, simplify.

a.$\left(4\sqrt{12z^3}\right)\left(3\sqrt{9z}\right)$

b.$\left(5\sqrt[3]{3x^3}\right)\left(3\sqrt[3]{18x^3}\right)$

Step-by-Step Solution

Answer

On multiplication, we get $72z^2\sqrt{3},45x^2\sqrt[3]{2}$  respectively

Part (a) Step 1: Given information

Given the expression $\left(4\sqrt{12z^3}\right)\left(3\sqrt{9z}\right),\left(5\sqrt[3]{3x^3}\right)\left(3\sqrt[3]{18x^3}\right)$

Part (a) Step 2: Multiplying the terms and simplifying

Multiplying the terms and simplifying, we get

$$\begin{gathered} \left(4\sqrt{12z^3}\right)\left(3\sqrt{9z}\right)=(4·3)\left(\sqrt{12z^3}·\sqrt{9z}\right) \\ \left(4\sqrt{12z^3}\right)\left(3\sqrt{9z}\right)=12·\sqrt{108z^4} \\ \left(4\sqrt{12z^3}\right)\left(3\sqrt{9z}\right)=12·\sqrt{36z^4·3} \\ \left(4\sqrt{12z^3}\right)\left(3\sqrt{9z}\right)=12·\sqrt{36z^4}·\sqrt{3} \\ \left(4\sqrt{12z^3}\right)\left(3\sqrt{9z}\right)=12·\sqrt{{\left(6z^2\right)}^2}·\sqrt{3} \\ \left(4\sqrt{12z^3}\right)\left(3\sqrt{9z}\right)=12·6z^2·\sqrt{3} \\ \left(4\sqrt{12z^3}\right)\left(3\sqrt{9z}\right)=72z^2\sqrt{3} \end{gathered}$$

Part (b) Step 1: Given information

Given expression$(-3\sqrt[4]{8})(-5\sqrt[4]{6})$

Part (b) Step 2: Multiplying the terms and simplifying

Simplifying, we get

$$\begin{gathered} \left(5\sqrt[3]{3x^3}\right)\left(3\sqrt[3]{18x^3}\right)=(5·3)\left(\sqrt[3]{3x^3}·\sqrt[3]{18x^3}\right) \\ \left(5\sqrt[3]{3x^3}\right)\left(3\sqrt[3]{18x^3}\right)=15·\sqrt[3]{54x^6} \\ \left(5\sqrt[3]{3x^3}\right)\left(3\sqrt[3]{18x^3}\right)=15·\sqrt[3]{27x^6·2} \\ \left(5\sqrt[3]{3x^3}\right)\left(3\sqrt[3]{18x^3}\right)=15·\sqrt[3]{27x^6}·\sqrt[3]{2} \\ \left(5\sqrt[3]{3x^3}\right)\left(3\sqrt[3]{18x^3}\right)=15·\sqrt[3]{{\left(3x^2\right)}^3}·\sqrt[3]{2} \\ \left(5\sqrt[3]{3x^3}\right)\left(3\sqrt[3]{18x^3}\right)=15·3x^2·\sqrt[3]{2} \\ \left(5\sqrt[3]{3x^3}\right)\left(3\sqrt[3]{18x^3}\right)=45x^2\sqrt[3]{2} \end{gathered}$$