What is the geometric mean of $7$ and $12$ ? $\newline$ (F) $1\frac{5}{7}$ $\newline$ (G) $9.5$ $\newline$ (H) $2\sqrt{21}$ $\newline$ (I) $4\sqrt{21}$

Full solution

Q. What is the geometric mean of

7

and

12

\newline

(F)

1\frac{5}{7}

\newline

(G)

9.5

\newline

(H)

2\sqrt{21}

\newline

(I)

4\sqrt{21}

Understand Geometric Mean: Understand the concept of geometric mean. The geometric mean of two numbers is the square root of their product. That is, if the numbers are $a$ and $b$ , their geometric mean is $\sqrt{a \times b}$ .
Calculate Product: Calculate the product of the two numbers. $\newline$ Product of $7$ and $12$ = $7 \times 12 = 84$
Find Square Root: Find the square root of the product to get the geometric mean. $\newline$ Geometric mean = $\sqrt{84}$
Simplify Square Root: Simplify the square root. $\newline$ $\sqrt{84}$ can be simplified by factoring out perfect squares. $\newline$ $84 = 4 \times 21$ , and $4$ is a perfect square. $\newline$ So, $\sqrt{84} = \sqrt{4 \times 21} = \sqrt{4} \times \sqrt{21} = 2\sqrt{21}$