Q. 8.90 - Intermediate Algebra

Question

Simplify-

(a) ${(6-\sqrt{5})}^2$

(b) ${(9-2\sqrt{10})}^2$

Step-by-Step Solution

Answer

The values are-

(a) $41-12\sqrt{5}$

(b) $121-36\sqrt{10}$

Step 1. Explanation

(a) and (b) We can identify the property of subtraction here, i.e., ${(a-b)}^2$.

We know that ${(a-b)}^2=a^2-2ab+b^2$.

By applying it, we will get the values.

Step 2. Solving for part (a)

We have ${(6-\sqrt{5})}^2$.

Applying the property of ${(a-b)}^2$ will give us,

$$\begin{gathered} ={(6-\sqrt{5})}^2 \\ ={\left(6\right)}^2-(2·6·\sqrt{5})+{\left(\sqrt{5}\right)}^2 \\ =36-12\sqrt{5}+5 \\ =41-12\sqrt{5} \end{gathered}$$

Thus we get the value as $41-12\sqrt{5}$.

Step 3. Solving for part (b)

We have ${(9-2\sqrt{10})}^2$.

Applying the property of ${(a-b)}^2$ will give us,

$$\begin{gathered} ={(9-2\sqrt{10})}^2 \\ ={\left(9\right)}^2-(2·9·2\sqrt{10})+{\left(2\sqrt{10}\right)}^2 \\ =81-36\sqrt{10}+40 \\ =121-36\sqrt{10} \end{gathered}$$
Thus we get the value as $121-36\sqrt{10}$.