Q. 208 - Intermediate Algebra

Question

In the following exercises, multiply:

$$\begin{gathered} \left(a\right) {\left(5-\sqrt{10}\right)}^2 \\ \left(b\right) {\left(8+3\sqrt{2}\right)}^2 \end{gathered}$$

Step-by-Step Solution

Answer

Part a. By multiplying we get, $35-10\sqrt{10}.$

Part b. By multiplying we get, $82+48\sqrt{2}.$

Part a Step 1. Multiply the given expression

The given expression is ${\left(5-\sqrt{10}\right)}^2.$

Multiply, using the Product of Binomial Squares Pattern

$$\begin{gathered} {\left(5-\sqrt{10}\right)}^2 \\ =5^2-2·5·\sqrt{10}-{\left(\sqrt{10}\right)}^2 \end{gathered}$$

Part a Step 2. Simplify

By simplifying we get,

$$\begin{gathered} =5^2-2·5·\sqrt{10}+{\left(\sqrt{10}\right)}^2 \\ =25-10\sqrt{10}+10 \\ =35-10\sqrt{10} \left[combine like term\right] \end{gathered}$$

Part b Step 1. Multiply the given expression

The given expression is ${\left(8+3\sqrt{2}\right)}^2.$

Multiply, using the Product of Binomial Squares Pattern

$$\begin{gathered} {\left(8+3\sqrt{2}\right)}^2 \\ =8^2+2·8·3\sqrt{2}+{\left(3\sqrt{2}\right)}^2 \end{gathered}$$

Part b Step 2. Simplify

By simplifying we get,

$$\begin{gathered} =8^2+2·8·3\sqrt{2}+{\left(3\sqrt{2}\right)}^2 \\ =64+48\sqrt{2}+18 \\ =82+48\sqrt{2} \left[combine like term\right] \end{gathered}$$