Simplify, giving solution with positive index : $(4 x^{2} y^{3})^{3} \div (3 x^{2} y^{3})^{3}$

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Solution

Given : $(4 x^{2} y^{3})^{3} \div (3 x^{2} y^{3})^{3}$
It can be written as:
$(4 x^{2} y^{3})^{3} \div (3 x^{2} y^{3})^{3} = \frac{4^{3} x^{2 \times 3} y^{3 \times 3}}{3^{3} x^{2 \times 3} y^{3 \times 3}}$
[ $∵$ $(a^{m})^{n} = a^{m n}$ ]

On further calculation , we have :
$(4 x^{2} y^{3})^{3} \div (3 x^{2} y^{3})^{3} = \frac{4^{3} x^{6} y^{9}}{3^{3} x^{6} y^{9}}$

$= \frac{4^{3} x^{6 - 6} y^{9 - 9}}{3^{3}}$

$= \frac{4^{3} x^{0} y^{0}}{3^{3}} = \frac{4^{3}}{3^{3}}$
$= \frac{64}{27}$ [ $∵$ $a^{0} = 1$ ]

Hence, $(4 x^{2} y^{3})^{3} \div (3 x^{2} y^{3})^{3} = \frac{64}{27}$

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