Exponents And Powers, Revision Notes: ICSE Class 8 MATHS, Mathematics

Exponents and Powers

Laws of rational exponents of real numbers:

Let a and b be two real numbers and m and n be two rational numbers then

- $a^{p} \cdot a^{q} = a^{p + q}$
- $(a^{p})^{q} = a^{p q}$
- $\frac{a^{p}}{a^{q}} = a^{p - q}$
- $a^{p} b^{p} = (a b)^{p}$
- $\frac{a^{n}}{b^{n}} = {(\frac{a}{b})}^{n}$
- $a^{- p} = \frac{1}{a^{p}}$

Example:

$\sqrt[3]{(512)^{- 2}} = [(512)^{- 2}]^{\frac{1}{3}} = (512)^{\frac{- 2}{3}} [(a^{m})^{n} = a^{m n}] = (8^{3})^{\frac{- 2}{3}} = (8)^{3 \times \frac{- 2}{3}} [(a^{m})^{n} = a^{m n}] = (8)^{- 2} = \frac{1}{8^{2}} [a^{- m} = \frac{1}{a^{m}}] = \frac{1}{64}$

If a natural number m can be expressed as n², where n is also a natural number, then m is a square number. The square numbers are also called perfect squares.

Examples: 1, 4, 9, 16, 25, 36 … are perfect squares.

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