## Surds and Indices Questions - 1

Value of $\displaystyle \left({125}\right)$$\displaystyle ^{\frac{2}{3}} = • 5 • 25 • 15 • 125 • 10 Explanation \displaystyle \left({5^3}\right)$$\displaystyle ^{\frac{2}{3}}$ = 5 $\displaystyle^{3 \times \frac{2}{3}} = 5^2$ = 25

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Value of $\displaystyle \left({36}\right)$$\displaystyle^{\frac{-3}{2}}$ =
• 6
• 36
• $\displaystyle \frac{\ \ 1}{216}$
• $\displaystyle \frac{1}{6}$
• $\displaystyle \frac{\ 2}{39}$
Explanation

$\displaystyle \left({36}\right)^{\frac{-3}{2}}$ = $\displaystyle \left({6^2}\right)^{\frac{-3}{2}} = \left(\frac{1}{6^2}\right)^{\frac{3}{2}} = \frac{1}{6^3} = \frac{1}{216}$

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Value of 312 x 36 =
• 33
• 34
• 312
• 318
• 38
Explanation

am x an = am+n

312 x 36 = 312+6 = 318

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Value of $\displaystyle \frac{2^8}{2^6}$ =
• 4
• 2
• 3
• 1
• 5
Explanation

$\displaystyle \frac{a^m}{a^n} = a^{m-n}$

$\displaystyle \frac{2^8}{2^6} = a^{8 - 6} = 2^2$ = 4.

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Value of $\displaystyle \left(3^2\right)^3$ =
• 243
• 27
• 2187
• 81
• 729
Explanation

$\displaystyle \left(a^m\right)^n = a^{mn}$

$\displaystyle \left(3^2\right)^3 = 3^{2 \times 3} = 3^6$ = 729

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Simplest form of $\displaystyle \sqrt{216}$ is
• $\displaystyle 2 \sqrt{108}$
• $\displaystyle 18 \sqrt{12}$
• $\displaystyle 3 \sqrt{18}$
• $\displaystyle 6 \sqrt{6}$
• $\displaystyle 6 \sqrt{12}$
Explanation

Simplified form of $\displaystyle \sqrt{216}$ is => $\displaystyle \sqrt{36 \times 6} => 6 \sqrt{6}$

Workspace
Value of $\displaystyle (\sqrt{27})^{\frac {1}{3}}$ is
• 9
• 3
• 6
• 81
• 36
Explanation

$\displaystyle (\sqrt{27})^{\frac {1}{3}}$ => $\displaystyle \sqrt[3]{27}$ = 3.

Workspace
Value of 2-6
• 16
• $\displaystyle \frac {\ 1}{16}$
• $\displaystyle \frac {\ 1}{64}$
• $\displaystyle \frac {1}{2}$
• $\displaystyle \frac {3}{2}$
Explanation

$\displaystyle 2^{-6} = \left(\frac {1}{2}\right)^6 = \frac{1}{64}$

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Entire form of $\displaystyle 3 \sqrt{15}$
• $\displaystyle \sqrt{45}$
• $\displaystyle \sqrt{135}$
• $\displaystyle \sqrt{125}$
• $\displaystyle \sqrt{35}$
• $\displaystyle \sqrt{65}$
Explanation

Entire form of $\displaystyle 3 \sqrt{15}$ is = $\displaystyle \sqrt{9 \times 15} = \sqrt{135}$

Workspace
$\displaystyle \sqrt[3]{64} = 2^n,$ n = ?
• 4
• 16
• 12
• 8
• 2
Explanation

$\displaystyle \sqrt[3]{64} = 2^n = 2^{6 \times \frac{1}{3}} = 2^2 = 2^n$ = n = 2.

Workspace

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## Surds and Indices Important Formulas

### Laws of Indices:

1. $\displaystyle a^m \times a^n = a^{m+n}$

2. $\displaystyle \frac{a^m}{a^n} = a^{m-n}$

3. $\displaystyle \left(ab\right) ^ n = a^n \times b^n$

4. $\displaystyle \left(a^m\right) ^ n = a^{mn}$

5. $\displaystyle \left(\frac {a}{b}\right) ^ n = \frac {a^n}{b^n}$

6. $\displaystyle a^0$ = 1

7. if $\displaystyle a^m = a^n$ = either m = o or a = b.

8. $\displaystyle a^m = a^n$, then a = 1 or m = n.

### Laws of Surds:

We write $\displaystyle \sqrt[n]{a} = a^{\frac {1}{n}}$ and it is called as a surd of order â€˜n'.

1. $\displaystyle \left(\sqrt[n]{a}\right) ^ n = \left({a^{\frac {1}{n}}}\right)^n = a^{\frac {1}{n} \times n} = a^{\frac {n}{n}}$ = a.

2. $\displaystyle \sqrt[n]{a}{b} = \sqrt[n]{a} \times \sqrt[n]{b}$

3. $\displaystyle \sqrt[n]\frac{a}{b} = \frac {\sqrt[n]{a}} {\sqrt[n]{b}}$

4. $\displaystyle \left(\sqrt[n]{a}\right) ^ m = \sqrt[n]{a^m}$

5. $\displaystyle \sqrt[m]{\sqrt[n]{{a}}} = \sqrt[mn]{a}$

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