# 7th Class Mathematics Exponents and Power Exponents and Powers

Exponents and Powers

Category : 7th Class

Exponents and Powers

• Exponential form is the short form of repeated multiplication. A number written in exponential form contains a base and an exponent.

${{10}^{5}}$is the exponential form of 1,00,000, since 1,00,000 =10$\times$10$\times$10$\times$10$\times$10.

In ${{10}^{5}},\,10$ is the base and 5 is the exponent or index or power.

• Base denotes the number to be multiplied and the power denotes the number of times the base is to be multiplied.

$a\times a={{a}^{2}}$(read as 'a squared' or 'a raised to the power 2')

$a\times a\times a={{a}^{3}}$(read as 'a cubed' or 'a raised to the power 3')

$a\times a\times a\times a={{a}^{4}}$ (read as 'a raised to the power 4' or ${{4}^{th}}$ power of a)

…………………………………………………….

$a\times a\times a\,....$ (n factors) $={{a}^{n}}$(read as 'a raised to the power n' or ${{\operatorname{n}}^{th}}$power of a)

• (i) When a negative number is raised to an even power the value is always positive.

e.g.,${{\left( -5 \right)}^{4}}=\left( -5 \right)\times \left( -5 \right)\times \left( -5 \right)\times \left( -5 \right)=+\,625$

• When a negative number is raised to an odd power, the value is always negative.

e.g., ${{\left( -3 \right)}^{5}}=\left( -3 \right)\times \left( -3 \right)\times \left( -3 \right)x\left( -3 \right)x\left( -3 \right)=\left( -\,243 \right)$

Note:    (a) ${{\left( -1 \right)}^{oddnumber}}=-1$

(b)${{\left( -1 \right)}^{oddnumber}}=+1$

• Laws of Exponents:

For any non-zero integers 'a' and V and whole numbers 'm' and 'n',

(i) $a\times a\times a\times ......\times a$(m factors)$={{a}^{m}}$

(ii) ${{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}$

(iii) $\frac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m+n}},\operatorname{if}\,\,m>n$

$=1,\text{ }if\text{ }m=n$

$=\frac{1}{{{a}^{n-m}}}\,\operatorname{if}\,m<n$

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

(v) ${{\left( ab \right)}^{m}}={{a}^{m}}{{b}^{m}}$

(vi) ${{\left( \frac{a}{b} \right)}^{m}}=\frac{{{a}^{m}}}{{{b}^{m}}}$

(vii)${{a}^{o}}=1$

• Any number can be expressed as a decimal number between 1.0 and 10.0 including 1.0 multiplied by a power of 10. Such a form of a number is called its standard form.

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