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

Workspace
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   

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

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

am x an = am+n

312 x 36 = 312+6 = 318

Workspace
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.

Workspace
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

Workspace
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   

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

Workspace
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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More Surds and Indices Practice Tests

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 ‘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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