CBSE Class 8CBSE
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Laws of Exponents

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notes

Example : `2^(-3) xx 2^(-2)`
`2^(-3) xx 2^(-2) = 1/2^3 xx 1/2^2`  

.... `[a^(-m) = 1/(a^m) "for any non-zero integer a"]`

`=1/(2^3 xx 2^2) = 1/(2^(3+2)) = 2^(-5)`

In general, we can say that for any non-zero integer a, `a^m × a^n = a^(m + n)`, where m and n are integers.
On the same lines you can verify the following laws of exponents, where a and b are non zero integers and m, n are any integers.

i) `(a^m)/(a^n) = a^(m-n)`

ii) `(a^m)^n = a^(mn)`

iii) `a^m xx b^m = (ab)^m`

iv) `(a^m)/(b^m) = (a/b)^m`

v) `a^0 = 1`

For Example : 

I) `(– 4)^5 × (– 4)^(–10) =  (– 4)^(5 – 10)`  
= `(– 4)^(–5) = 1/(-4)^5 `  ....  `(a^m × a^n = a^(m + n), a^(-m) = 1/(a^m) )` 

II) Express `4^(–3)` as a power with the base 2.
`4^(–3) = (2 xx 2)^(-3) = (2^2)^(-3) = 2^(2 xx -3) = 2^(-6)`  ....`[(a^m)^n = a^(mn)]`

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