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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:

\(\sqrt{2} \times \sqrt{2} = 2\), \(\sqrt{3} \times \sqrt{3} = 3\)

Did you find an error?