7th Class Mathematics Exponents and Power Exponents

Exponents

Exponents

The continued product of a number multiplied with itself a number of times can be written in exponent form as \[{{a}^{n}}\], where 'n' is a natural number and 'a' is any number.

i.e. \[{{a}^{n}}=a\times a\times a\]..... up to n times. Here a is the base and n is exponent (or index or power).

For any rational number\[\left( \frac{p}{q} \right),{{\left( \frac{p}{q} \right)}^{n}}=\frac{p}{q}\times \frac{p}{q}\times \frac{p}{q}\times \].......... up to n times

Laws of Exponents

The following are the laws of exponent:

\[\Rightarrow \]\[{{X}^{m}}\times {{X}^{n}}={{X}^{m+n}}\]            \[\Rightarrow \]\[\frac{{{X}^{m}}}{{{X}^{n}}}={{X}^{m-n}}\]

\[\Rightarrow \]\[{{X}^{m}}\times {{Y}^{m}}={{(X\times Y)}^{m}}\] \[\Rightarrow \]\[{{\left[ {{\left( \frac{X}{Y} \right)}^{n}} \right]}^{m}}={{\left( \frac{X}{Y} \right)}^{mn}}\]

\[\Rightarrow \]\[{{\left( \frac{X}{Y} \right)}^{-n}}={{\left( \frac{Y}{X} \right)}^{n}}\]                 \[\Rightarrow \]\[{{X}^{0}}=1\]

\[\Rightarrow \]\[{{X}^{1}}=X\] \[\Rightarrow \]\[{{X}^{-1}}=\frac{1}{X}\]

Uses of Exponents

It is the way to represent the smaller as well as larger numbers which are not possible to write in the convenient way by existing number system. Suppose there are one hundred billion stars in a galaxy. It is written as 100000000000 which is not easy to write or read but in the exponents form it can be written as \[{{10}^{11}}\]. The size of the microbes or atoms is very-very small but it can be easily expressed in exponential form.

Example:

1. Simplify: \[{{(4)}^{3}}+{{(3)}^{2}}\]

(a) \[-125\]                     (b) 73

(c) 64                            (d) 576

(e) None of these

Answer (b)

Explanation: \[{{(4)}^{3}}=4\times 4\times 4=64\] and \[{{(3)}^{2}}=3\times 3=9\]

Now, \[{{(4)}^{3}}+{{(3)}^{2}}=64+9=73\]

Example:

Values of \[{{2}^{5}}\]and \[{{2}^{-5}}\] are respectively:

(a) \[\frac{1}{32}\,\]and 32                     (b) 16 and \[\frac{1}{16}\,\]

(c) 32 and \[\frac{1}{32}\,\]                    (d) \[\frac{1}{16}\,\] and 16

(e) None of these

Answer (c)

Explanation: \[{{2}^{5}}=2\times 2\times 2\times 2\times 2=32\]

and \[{{2}^{-5}}={{\left( \frac{1}{2} \right)}^{5}}=\frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}=\frac{1}{32}\]

Example:

By what number \[{{\left( \frac{1}{3} \right)}^{-5}}\] must be multiplied so that the result is  \[\frac{2}{3}\].

(a) \[\frac{2}{243}\]                   (b) \[\frac{2}{729}\]

(c) \[\frac{2}{81}\]                     (d) \[\frac{2}{27}\]

(e) None of these

Answer (b)

Explanation: Required number \[=\frac{2}{3}\div {{\left( \frac{1}{3} \right)}^{-5}}=\frac{2}{3}\times {{3}^{-5}}\]

\[=\frac{2}{3}\times \frac{{{1}^{5}}}{3}=\frac{2}{3}\times \frac{1}{3}\times \frac{1}{3}\times \frac{1}{3}\times \frac{1}{3}\times \frac{1}{3}=\frac{2}{729}\]