PUMPA - THE SMART LEARNING APP

AI system creates personalised training plan based on your mistakes

For any positive numbers $$a$$ and $$b$$, we have:

1. $\sqrt{\mathit{ab}}=\sqrt{a×b}=\sqrt{a}×\sqrt{b}$

2. $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$
Example:
1. Find the value of $\sqrt{81×36}$.

Solution:

We need to find the value of $\sqrt{81×36}$.

Apply the product rule of square root.

$\sqrt{\mathit{ab}}=\sqrt{a×b}=\sqrt{a}×\sqrt{b}$

$\sqrt{81×36}=\sqrt{81}×\sqrt{36}$

$=\sqrt{{9}^{2}}×\sqrt{{6}^{2}}$

$=9×6$  (square and square root get cancelled.)

$$=$$ $$54$$

Therefore, the value of$\sqrt{81×36}$ $$=$$ $$54$$.

2. Simplify: $\sqrt{\frac{48}{75}}$

Solution:

$\sqrt{\frac{48}{75}}=\sqrt{\frac{16×3}{25×3}}$

Now, cancel the common factor $$3$$.

$\sqrt{\frac{48}{75}}=\sqrt{\frac{16}{25}}$

Apply the quotient rule, $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$, $$b \ne 0$$.

$\sqrt{\frac{48}{75}}=\frac{\sqrt{16}}{\sqrt{25}}$

$\sqrt{\frac{48}{75}}=\frac{4}{5}$

Therefore, $\frac{4}{5}$ is the simplified value of $\sqrt{\frac{48}{75}}$.

3. Find the value of $\sqrt{10.24}$ by applying the quotient rule.

Solution:
We can write $\sqrt{10.24}$ $$=$$ $\sqrt{\frac{1024}{100}}$.

Now, apply the quotient rule $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$, $$b \ne 0$$.

$\sqrt{10.24}$ $=\frac{\sqrt{1024}}{\sqrt{100}}$

$=\frac{\sqrt{{32}^{2}}}{\sqrt{{10}^{2}}}$

$=\frac{32}{10}$

$$=$$ $$3.2$$

Therefore, $\sqrt{10.24}$ $$=$$ $$3.2$$.
Multiplying identical square root numbers
If we multiply the square root number by itself, we get the same number without the square root.

$\sqrt{a\phantom{\rule{0.147em}{0ex}}}\phantom{\rule{0.147em}{0ex}}×\sqrt{a}\phantom{\rule{0.147em}{0ex}}=a$
Example:
$$\sqrt{2} \times \sqrt{2} = 2$$, $$\sqrt{3} \times \sqrt{3} = 3$$