Q. 198 - Intermediate Algebra

Question

In the following exercises, multiply.

a.$(3-2\sqrt{7})(5-4\sqrt{7})$

b.$(\sqrt[3]{x}-5)(\sqrt[3]{x}-3)$

Step-by-Step Solution

Answer

On multiplication, we get $71-22\sqrt{7},\sqrt[3]{x^2}-8\sqrt[3]{x^2}+15$ respectively

Part (a) Step 1: Given information

Given the expression $(3-2\sqrt{7})(5-4\sqrt{7})$

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

Multiplying and simplifying we get

$$\begin{gathered} (3-2\sqrt{7})(5-4\sqrt{7})=15-12\sqrt{7}-10\sqrt{7}+8\sqrt{(7{)}^2} \\ (3-2\sqrt{7})(5-4\sqrt{7})=15-12\sqrt{7}-10\sqrt{7}+8·7 \\ (3-2\sqrt{7})(5-4\sqrt{7})=(15+56)-(12\sqrt{7}+10\sqrt{7}) \\ (3-2\sqrt{7})(5-4\sqrt{7})=71-22\sqrt{7} \end{gathered}$$

Part (b) Step 1: Given information

Given expression $(\sqrt[3]{x}-5)(\sqrt[3]{x}-3)$

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

Multiplying the terms and simplifying, we get

$$\begin{gathered} (\sqrt[3]{x}-5)(\sqrt[3]{x}-3)=\sqrt[3]{x^2}-3\sqrt[3]{x}-5\sqrt[3]{x}+15 \\ (\sqrt[3]{x}-5)(\sqrt[3]{x}-3)=\sqrt[3]{x^2}(-3\sqrt[3]{x}-5\sqrt[3]{x})+15 \\ (\sqrt[3]{x}-5)(\sqrt[3]{x}-3)=\sqrt[3]{x^2}-8\sqrt[3]{x^2}+15 \\ (\sqrt[3]{x}-5)(\sqrt[3]{x}-3)=-7\sqrt[3]{x^2}+15 \end{gathered}$$