7th Class Mathematics Exponents and Power Exponents and Powers

 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.