The problem asks us to calculate the values of four expressions: a) $\sqrt{81}$ b) $\sqrt{3^2}$ c) $\sqrt{(\frac{2}{3})^2}$ d) $\sqrt{7} \times \sqrt{3}$

1. Problem Description

The problem asks us to calculate the values of four expressions:

a) $\sqrt{81}$

b) $\sqrt{3^2}$

c) $\sqrt{(\frac{2}{3})^2}$

d) $\sqrt{7} \times \sqrt{3}$

2. Solution Steps

a) We need to find the square root of

8

1. Since $9 \times 9 = 81$, we have $\sqrt{81} = 9$.

b) We need to find the square root of $3^2$.

$3^2 = 3 \times 3 = 9$, so $\sqrt{3^2} = \sqrt{9}$.

Since $3 \times 3 = 9$, we have $\sqrt{9} = 3$.

Alternatively, $\sqrt{a^2} = |a|$. So, $\sqrt{3^2} = |3| = 3$.

c) We need to find the square root of $(\frac{2}{3})^2$.

$(\frac{2}{3})^2 = \frac{2}{3} \times \frac{2}{3} = \frac{4}{9}$, so $\sqrt{(\frac{2}{3})^2} = \sqrt{\frac{4}{9}}$.

Since $\frac{2}{3} \times \frac{2}{3} = \frac{4}{9}$, we have $\sqrt{\frac{4}{9}} = \frac{2}{3}$.

Alternatively, $\sqrt{a^2} = |a|$. So, $\sqrt{(\frac{2}{3})^2} = |\frac{2}{3}| = \frac{2}{3}$.

d) We need to find the product of $\sqrt{7}$ and $\sqrt{3}$.

$\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}$

So, $\sqrt{7} \times \sqrt{3} = \sqrt{7 \times 3} = \sqrt{21}$.

3. Final Answer

a) 9

b) 3

c) $\frac{2}{3}$

d) $\sqrt{21}$