Laws of exponents — lesson. Mathematics State Board, Class 8.

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Product law

The product law states that the exponents can be added when multiplying two powers with the same base.

$a^{m} \times a^{n} = a^{m + n}$ , where

$a \ne 0$ and

$a$ ,

$m$ ,

$n$ are integers.

Example:

$3^4 \times 3^2$

Here, the base

$3$ is the same for both the numbers. So, we can add the powers.

$a^{m} \times a^{n} = a^{m + n}$

$3^4 \times 3^2 = 3^{4 + 2} = 3^{6}$

$5^{-4} \times 5^{-2}$

Method I:

$5^{-4} \times 5^{-2}$

$=$

$\frac{1}{5^{4}} \times \frac{1}{5^{2}}$

$=$

$\frac{1}{5^{4} \times 5^{2}}$

$=$

$\frac{1}{5^{4 + 2}}$

$=$

$\frac{1}{5^{6}}$

$=$

$5^{-6}$

Thus,

$5^{-4} \times 5^{-2}$

$=$

$5^{-6}$ .

Method II:

$5^{-4} \times 5^{-2}$

$=$

$5^{(-4)+(-2)} = 5^{-6}$

Quotient law

The quotient law states that we can divide two powers with the same base by subtracting the exponents.

$\frac{a^{m}}{a^{n}} = a^{m - n}$ , where

$a \ne 0$ and

$a$ ,

$m$ ,

$n$ are integers.

Example:

$\frac{5^{4}}{5^{2}} = 5^{4 - 2} = 5^{2}$

$\frac{{(- 4)}^{6}}{{(- 4)}^{- 2}} = {(- 4)}^{6 - (- 2)}$

$=$

$(-4)^{6+2}$

$=$

$(-4)^{8}$

Therefore,

$\frac{{(- 4)}^{6}}{{(- 4)}^{- 2}} = {(- 4)}^{8}$ .

Power law

The power law states that when a number is raised to a power of another power, we need to multiply the powers or exponents.

$(a^m)^n = a^{mn}$ , where

$a \ne 0$ and

$a$ ,

$m$ ,

$n$ are integers.

Example:

$(5^2)^3 = (5)^{2 \times 3} = 5^{6}$ .

$[5^{(-2)}]^3 = 5^{(-2) \times 3} = (5)^{-6}$ .

$[(-5)^{2}]^{3} = (-5)^{2 \times 3} = (-5)^{6}$ .

$[(-5)^{2}]^{-3} = (-5)^{2 \times (-3)} = (-5)^{-6}$ .

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4. Laws of exponents

Theory: