Question

Simplify Using laws of Exponents.
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Answer 1

Required Solution:-

$:\implies\tt\dfrac{3^{-4}\times15^{-3}\times 625}{3^{-7}\times6^{-2}}$

$:\implies\tt\dfrac{3^{-4}\times(5\times 3)^{-3}\times 625}{3^{-7}\times6^{-2}}$

$\bf As\: we\; know\: that\: (ab)^{-2}=a^{-2}\times b^{-2} \:where\;a=3\:b=5$

So,

$:\implies\tt\dfrac{3^{-4}\times3^{-3}\times 5^{-3}\times 625}{3^{-7}\times6^{-2}}$

$:\implies\tt\dfrac{3^{-4}\times3^{-3}\times 5^{-3}\times 5^4}{3^{-7}\times(3\times 2)^{-2}}$

$\bf As\: we\; know\: that\: (ab)^{-2}=a^{-2}\times b^{-2} \:where\;a=3\:b=2$

So,

$:\implies\tt\dfrac{3^{-4}\times3^{-3}\times 5^{-3}\times 5^4}{3^{-7}\times3^{-2}\times 2^{-2}}$

$\bf Now\:we\: know\: that\:a^x\times a^y=a^{(x+y)}$

So,

$:\implies\tt\dfrac{3^{-4+(-3)}\times 5^{4+(-3)}}{3^{-7+(-2)}\times2^{-2}}$

$:\implies\tt\dfrac{3^{-7}\times 5^{1}}{3^{-9}\times2^{-2}}$

$\bf Now\;we\: know\; a^y\div a^x=a^{y-x}$

So,

$:\implies\tt\dfrac{3^{-7-(-9)}\times 5}{2^{-2}}$

$:\implies\tt\dfrac{3^{(-7+9)}\times 5}{2^{-2}}$

$:\implies\tt\dfrac{3^2\times 5}{2^{-2}}$

Transporting $\bf 2^{-2}\:from\:Dn.\:to\:Nr.\:will\:make\:its\: power\:+ve$

$:\implies\tt3^2\times 5\times 2^2$

Now simplifying it:

$:\implies\tt(9\times2) \times(5\times 2)$

$:\implies\tt(18) \times(10)$

$\Large\boxed{:\implies\tt 180}$

So the final answer is 180.

Laws of exponents:

$\boxed{\begin{array}{l}\bf{\dag}\:\:\underline{\text{Law of Exponents :}}\\\\\bigstar\:\:\sf\dfrac{a^m}{a^n} = a^{m - n}\\\\\bigstar\:\:\sf{(a^m)^n = a^{mn}}\\\\\bigstar\:\:\sf(a^m)(a^n) = a^{m + n}\\\\\bigstar\:\:\sf\dfrac{1}{a^n} = a^{-n}\\\bigstar\:\:\sf\sqrt[\sf n]{\sf a} = (a)^{\dfrac{1}{n}}\end{array}}$

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