Question

Simplify, giving solution with positive index​ : (xxi) $\frac{\left(5x^7{)}^3\right(10x^2{)}^2}{(2x^6{)}^7}$

Answer 1

Given, $\frac{\left(5x^7{)}^3\right(10x^2{)}^2}{(2x^6{)}^7}$
It can be written as:

$\frac{\left(5x^7{)}^3\right(10x^2{)}^2}{(2x^6{)}^7}=\frac{5^3{x}^{21}{10}^2{x}^4}{2^7{x}^{42}}$

$=\frac{5^3\times (2\times 5{)}^2\times x^{21+4}}{2^7{x}^{42}}$
[$\because$$a^m\times a^n=a^{m+n}$]

$=\frac{5\times 5\times 5\times 2\times 5\times 2\times 5\times x^{25-42}}{2\times 2\times 2\times 2\times 2\times 2\times 2}$

$=\frac{5^5\times x^{-17}}{2^5}$

$=\frac{3125}{32x^{17}}$

Hence, $\frac{\left(5x^7{)}^3\right(10x^2{)}^2}{(2x^6{)}^7}=\frac{3125}{32x^{17}}$