Power, Indices and Surds

Pratice Power, Indices and Surds Questions and answers.

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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.