# Power, Indices and Surds

Pratice Power, Indices and Surds Questions and answers.

Rule Description
1

If any number is multiplied by the same number ‘n’ times, then,

$a×a×a× ............. × a \ (n \ times ) = a^n$

1. where n and a are real numbers.
2. a is called base.
3. n is called indices.
2
• $a^m\times a^n=a^{m+n}$
• $a^m\times a^n\times a^p = a^{m+n+p}$
3 $a^x\times b^x\times c^x=\left(abc\right)^x$
4 $a^m\div a^n=a^{m-n}$
5
• $a^{-m} =\frac{1}{a^{m}}$
• $a^m=\frac{1}{a^{-m}}$
6
• $\left(a^m\right)^{^n}=a^{mn}$
• $\left(a^m\right)^{\frac{1}{n}}=a^{\frac{m}{n}}$
• $\left\{\left(a^m\right)^{^n}\right\}^{^{^p}}=a^{mnp}$
7
• $a^{m^n}\ne\left(a^m\right)^{^n}$
• $\left(a^m\right)^{\frac{1}{n}}=a^{\frac{m}{n}}$
• $a^{m^{n^p}}\ne\left\{\left(a^m\right)^{^n}\right\}^{^{^p}}$
8
• $\left(\frac{a}{b}\right)^{m}=\frac{a^m}{b^m}$
• $\left(\frac{a}{b}\right)^{-m}=\left(\frac{b}{a}\right)^{^m}$
9
• If $a^x=a^y$ then $x=y$
• If $a^x=a^y$ then $x=y$
10

If the indices on any number is zero, the value of that number is 1, as

$x^0 = 1$, $5^0 = 1$, $(5000)^0 = 1$

11 $\sqrt[n]{a}=\left(a\right)^{\frac{1}{n}}$
12 $\left(\sqrt[n]{a}\right)^n=a$
13 $\sqrt[n]{ab}=\sqrt[n]{a}\times\sqrt[n]{b}=\left(a\right)^{\frac{1}{n}}\times\left(b\right)^{\frac{1}{n}}$
14 $\sqrt[n]{\sqrt[n]{a}}=\left(\left(a\right)^{\frac{1}{n}}\right)^{\frac{1}{n}}=a^{n^{\frac{1}{2}}}$
15 $n\sqrt{\frac{a}{b}}=\frac{n\sqrt{a}}{n\sqrt{b}}=\left(\frac{a}{b}\right)^{\frac{1}{n}}$
16 $\sqrt[m]{\sqrt[n]{a}}=\sqrt[mn]{a}$
17 $\sqrt{x\sqrt{x\sqrt{x\sqrt{x..........n\ times}}}}=x^{\left(1-\frac{1}{x^n}\right)}$
18

If $x=n(n+1)$,

then$\sqrt{x-\sqrt{x-\sqrt{x-.....\infty}}}=n$

19

If $x=n(n+1)$,

then$\sqrt{x+\sqrt{x+\sqrt{x+.....\infty}}}=(n+1)$

20

$\sqrt[a]{b}, \sqrt[c]{d}, \sqrt[e]{f}, \sqrt[g]{h}$

To find smallest or greatest out of these, we should equate all the indices and compare the base.