**Question:**

Find the GM between the numbers

(i) 5 and 125

(ii) 1 and $\frac{9}{16}$

(iii) $0.15$ and $0.0015$

(iv) $-8$ and $-2$

(v) $-6.3$ and $-2.8$

(vi) $\mathrm{ad} \mathrm{ab}^{3}$

**Solution:**

(i) 5 and 125

To find: Geometric Mean

Given: The numbers are 5 and 125

Formula used: (i) Geometric mean between $a$ and $b=\sqrt{a b}$

Geometric mean of two numbers $=\sqrt{a b}$

$=\sqrt{5 \times 25}$

$=\sqrt{625}$

= 25

The geometric mean between 5 and 125 is 25

(ii) 1 and $\frac{9}{16}$

To find: Geometric Mean

Given: The numbers are 1 and $\frac{9}{16}$

Formula used: (i) Geometric mean between $a$ and $b=\sqrt{a b}$

Geometric mean of two numbers $=\sqrt{a b}$

$=\sqrt{1 \times \frac{9}{16}}$

$=\sqrt{\frac{9}{16}}$

$=\frac{3}{4}$

The geometric mean between 1 and $\frac{9}{16}$ is $\frac{3}{4}$.

(iii) 0.15 and 0.0015

To find: Geometric Mean

Given: The numbers are 0.15 and 0.0015

Formula used: (i) Geometric mean between $a$ and $b=\sqrt{a b}$

Geometric mean of two numbers $=\sqrt{a b}$

$=\sqrt{0.15 \times 0.0015}$

$=\sqrt{0.000225}$

= 0.015

The geometric mean between 0.15 and 0.0015 is 0.015.

(iv) -8 and -2

To find: Geometric Mean

Given: The numbers are -8 and -2

Formula used: (i) Geometric mean between $a$ and $b=\sqrt{a b}$+

Geometric mean of two numbers $=\sqrt{a b}$

$=\sqrt{-8 \times-2}$

$=\sqrt{16}$

$=\pm 4$

Mean is a number which has to fall between two numbers.

Therefore we will take $-4$ as our answer as $+4$ doesn't lie between $-8$ and $-2$

The geometric mean between $-8$ and $-2$ is $-4$.

(v) $-6.3$ and $-2.8$

To find: Geometric Mean

Given: The numbers are -6.3 and -2.8

Formula used: (i) Geometric mean between a and $b=\sqrt{a b}$

Geometric mean of two numbers $=\sqrt{a b}$

$=\sqrt{-6.3 \times-2.8}$

$=\sqrt{17.64}$

$=\pm 4.2$

Mean is a number which has to fall between two numbers.

Therefore we will take -4.2 as our answer as +4.2 doesn’t lie between -6.3 and -2.8

The geometric mean between -6.3 and -2.8 is -4.2.

(vi) $a^{3} b$ and $a b^{3}$

To find: Geometric Mean

Given: The numbers are $\mathrm{a}^{3} \mathrm{~b}$ and $\mathrm{ab}^{3}$

Formula used: (i) Geometric mean between $a$ and $b=\sqrt{a b}$

Geometric mean of two numbers $=\sqrt{a b}$

$=\sqrt{a^{3} b \times a b^{3}}$

$=\sqrt{a^{4} b^{4}}$

$=a^{2} b^{2}$

The geometric mean between $\mathrm{a}^{3} \mathrm{~b}$ and $\mathrm{ab}^{3}$ is $\mathrm{a}^{2} \mathrm{~b}^{2}$.