If 5^m 5^3 5^ -2 5^ -5 =5^ 12 , then find m. - Vidyadip

Question

If $\frac{5^\text{m}\times5^3\times5^{-2}}{5^{-5}}=5^{12}$, then find m.

✓

Answer

Given,
$\frac{5^\text{m}\times5^3\times5^{-2}}{5^{-5}}=5^{12}$
Using laws of exponents, $a^m÷ a^n= (a)^{m-n}$ and $\text{a}^{-\text{m}}=\frac{1}{\text{a}^\text{n}}$ [$\because$ a is non-zero integer]
Then,
$5^m\times 5^3\times 5^{-2}\times 5^5 = 5^{12}$
$\Rightarrow 5^m\times 5^8\times 5^{-2}= 5^{12}$
$\Rightarrow 5^m\times 5^6= 5^{12}$
$\Rightarrow 5^{m+6}= 5^{12}[\because a^m\times a^n= a^{m+n}]$
On camparing both sides, we get
$m + 6 = 12$
$\Rightarrow m = 6$

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