Exponents and Powers-Notes

•   The numbers with negative exponents also obey the following laws:

        (i) x^m × xⁿ = x^{m + n}

(ii) x^m ÷ xⁿ = x^{m – n}

        (iii) x^m × b^m = (x^b)^m

(ii) x⁰ = 1

        

•   A number is said to be in the standard Form, if it is expressed as the product of a number between 1 and 10 and the integral power of 10.

•   Very small numbers can be expressed in standard form using negative exponents.

When we write 54, it means 5 × 5 × 5 × 5, i.e. 5 is multiplied 4 times. So 5 is the base and 4 is the exponent. We read 54 as �5 raised to the power 4�.

We also know the following laws of exponents

        (i) x^m × xⁿ = x^{m + n}

(ii) x^m ÷ xⁿ = x^{m – n}

        (iii) x^m)ⁿ = x^{m × n}

(iv) x^m × y^m = (xy)^m

        

        The value of any number raised to 0 is 1, i.e. a⁰ = 1.

        We express very small or very large numbers in standard form (i.e scientific notation) for example:

        

For a non-zero integer x, we have

                

or x^�m × x^m = 1

So x^�m is the reciprocal (or the multiplicative inverse) of x^m and vice versa.

For example: (i) Reciprocal of 8^�7 = 8⁷ and

                                (ii) Reciprocal of 8⁷ = 8^�7