CBSE Class 8 Mathematics Exponents and Powers Notes Set 07

Negative Exponents

For a non-zero number a and a positive integer n, \( a^{-n}=\frac{1}{a^n} \). So \( 3^{-2}=\frac{1}{9} \) and \( \left(\frac{2}{3}\right)^{-2}=\left(\frac{3}{2}\right)^2=\frac{9}{4} \). The number \( a^{-n} \) is the reciprocal of \( a^n \).

Laws of Exponents

For non-zero a and b and integers m and n: \( a^m\times a^n=a^{m+n} \); \( a^m\div a^n=a^{m-n} \); \( (a^m)^n=a^{mn} \); \( a^m\times b^m=(ab)^m \); \( \frac{a^m}{b^m}=\left(\frac{a}{b}\right)^m \); \( a^0=1 \).

Using the Laws

To find the number by which a^m should be multiplied to get a^n, divide: the number is a^(n-m). To solve an equation such as 5^m ÷ 5^(-3) = 5^5, reduce both sides to the same base and equate the exponents: m + 3 = 5 so m = 2.

Standard Form (Scientific Notation)

A very large or very small number can be written as k x 10^n with 1 <= k < 10 and n an integer. Move the decimal point until one non-zero digit is before it; moving to the left gives a positive n and moving to the right gives a negative n.
Examples: 0.000007 = 7 x 10^(-6) and 150,000,000,000 = 1.5 x 10^11. To add numbers in standard form, first write them with the same power of 10.

Worked Example

Simplify (3^(-1) + 4^(-1) + 5^(-1))^0.
Any non-zero number raised to the power 0 equals 1, so the value is 1.

Worked Example

Express 0.00001275 m in standard form.
Moving the decimal point 5 places to the right gives 1.275, so the number is 1.275 x 10^(-5) m.