Q. 206 - Intermediate Algebra

Question

In the following exercises, multiply: 

$$\begin{gathered} \left(a\right) {\left(4+\sqrt{11}\right)}^2 \\ \left(b\right) {\left(3-2\sqrt{5}\right)}^2 \end{gathered}$$

Step-by-Step Solution

Answer

Part a. By multiplying we get, $27+8\sqrt{11}.$

Part b. By multiplying we get, $29-12\sqrt{5}.$

Part (a) Step 1. Multiply the given expression

The given expression is ${\left(4+\sqrt{11}\right)}^2$

Multiply, using the Product of Binomial Squares Pattern

$$\begin{gathered} {\left(4+\sqrt{11}\right)}^2 \\ =4^2+2·4·\sqrt{11}+{\left(\sqrt{11}\right)}^2 \end{gathered}$$

Part (a) Step 2. Simplify

By simplifying we get,

$$\begin{gathered} =4^2+2·4·\sqrt{11}+{\left(\sqrt{11}\right)}^2 \\ =16+8\sqrt{11}+11 \\ =27+8\sqrt{11} \left[combine like terms\right] \end{gathered}$$

Part (b) Step 1. Multiply the given expression

The given expression is ${\left(3-2\sqrt{5}\right)}^2.$

Multiply, using the Product of Binomial Squares Pattern

$$\begin{gathered} {\left(3-2\sqrt{5}\right)}^2 \\ =3^2-2·3·2\sqrt{5}+{\left(2\sqrt{5}\right)}^2 \end{gathered}$$

Part (b) Step 2. Simplify

By simplifying we get,

$$\begin{gathered} =3^2-2·3·2\sqrt{5}+{\left(2\sqrt{5}\right)}^2 \\ =9-12\sqrt{5}+20 \\ =29-12\sqrt{5} \end{gathered}$$