Q. 200 - Intermediate Algebra

Question

In the following exercises, multiply. 

a.$(7-2\sqrt{5})(4+9\sqrt{5})$

b.$(3\sqrt[3]{x}+2)(\sqrt[3]{x}-2)$

Step-by-Step Solution

Answer

On multiplication, we get $-62+55\sqrt{5},\sqrt[3]{3x^2}-4\sqrt[3]{x}-4$ respectively

Part (a) Step 1: Given information

Given the expression $(7-2\sqrt{5})(4+9\sqrt{5})$

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

Multiplying and simplifying, we get

$$\begin{gathered} (7-2\sqrt{5})(4+9\sqrt{5})=28+63\sqrt{5}-8\sqrt{5}-18\sqrt{(5{)}^2} \\ (7-2\sqrt{5})(4+9\sqrt{5})=28+63\sqrt{5}-8\sqrt{5}-18·5 \\ (7-2\sqrt{5})(4+9\sqrt{5})=(28-90)(63\sqrt{5}-8\sqrt{5}) \\ (7-2\sqrt{5})(4+9\sqrt{5})=-62+55\sqrt{5} \end{gathered}$$

Part (b) Step 1: Given information

Given expression $(3\sqrt[3]{x}+2)(\sqrt[3]{x}-2)$

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

Multiplying the terms and simplifying, we get

$$\begin{gathered} (3\sqrt[3]{x}+2)(\sqrt[3]{x}-2)=3\sqrt[3]{x^2}-6\sqrt[3]{x}+2\sqrt[3]{x}-4 \\ (3\sqrt[3]{x}+2)(\sqrt[3]{x}-2)=3\sqrt[3]{x^2}(-6\sqrt[3]{x}+2\sqrt[3]{x})-4 \\ (3\sqrt[3]{x}+2)(\sqrt[3]{x}-2)=3\sqrt[3]{x^2}-4\sqrt[3]{x}-4 \end{gathered}$$