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Concept Detail
Maths / Exponents and Power / Law 1 Product of exponents
(A Brief Glimpse of ABHYAS Content - Have aLook !!!!)

## Law 1 Product of exponents

### Multiplication with the same base

If we have to multiply the powers which have the same base then we have to add the exponents i.e. $a^m \times a^n = a^{m+n}$

Example:

• $8^3\;\;X \;\;8^4 = 8^{3+4} = 8^7$
• $\left ( \frac{2}{5} \right )^{5}\;\;X \;\;\left (\frac{2}{5} \right )^8 =\left (\frac{2}{5} \right )^{13}$
• Some Important Points to Remember:

• $(-1)^{odd \; number} = (-1)$
• $(-1)^{even \; number} = 1$
• $a^3b^2 \neq a^2b^3$
• $a^2b^3 = b^3a^2$
• Illustration:  Evaluate $\left ( 2 \right )^{7}\times \left ( 2 \right )^{13}$.

Solution: By the first law of exponents, we have $a^m \times a^n = a^{m+n}$

Hence $\left ( 2 \right )^{7}\times \left ( 2 \right )^{13}=\left ( 2 \right )^{20}$

Illustration:  Evaluate $\left ( -3 \right )^{7}\times \left ( -3\right )^{4}$.

Solution: By the first law of exponents, we have $a^m \times a^n = a^{m+n}$

Also we know that $(-1)^{odd\; number}=-1\;\;and\;\;(-1)^{even\; number}=1$

Hence we have, $\left ( -3 \right )^{7}\times \left ( -3\right )^{4}=\left ( -3 \right )^{11}$$=-(3)^{11}$

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