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7. Square root of product and quotient of numbers

For any positive numbers

$a$ and

$b$ , we have:

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

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

Example:

1. Find the value of

$\sqrt{81 \times 36}$ .

Solution:

We need to find the value of

$\sqrt{81 \times 36}$ .

Apply the product rule of square root.

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

$\sqrt{81 \times 36} = \sqrt{81} \times \sqrt{36}$

$= \sqrt{9^{2}} \times \sqrt{6^{2}}$

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

$=$

$54$

Therefore, the value of

$\sqrt{81 \times 36}$

$=$

$54$ .

2. Simplify:

$\sqrt{\frac{48}{75}}$

Solution:

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

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.

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

Example:

$\sqrt{2} \times \sqrt{2} = 2$ ,

$\sqrt{3} \times \sqrt{3} = 3$

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7. Square root of product and quotient of numbers

Theory: