Question

((2/3)^6)* ((9/4)^5)=(3/2)^(m+2) find the value of m​

Answer 1

$\sf\huge\bold{\underline{\underline{{Solution}}}}$

$\sf \large {( \frac{2}{3} )}^{6} \times {( \frac{9}{4}) }^{5} = ({ \frac{3}{2} )}^{m + 2}$

$\sf \large\longmapsto { (\frac{2}{3} )}^{6} \times ( { ({ \frac{3}{2}) }^{2}) }^{5} = { (\frac{3}{2} )}^{m + 2}$

$\sf \large \longmapsto {( \frac{2}{3} )}^{6} \times { (\frac{3}{2} )}^{10} = {( \frac{3}{2}) }^{m + 2}$

$\sf \large\longmapsto {( \frac{2}{3} )}^{6} = { (\frac{3}{2} )}^{m + 2} \div { (\frac{3}{2}) }^{10}$

Concept

When base are same power gets added in multiplication and gets substracted during division.

$\sf \large \frac{ {a}^{m} }{ {a}^{n} } = {a}^{m - n}$

$\sf \large {a}^{m} \times {a}^{n} = {a}^{m + n}$

$\sf \large \longmapsto { (\frac{2}{3}) }^{6} = {( \frac{3}{2}) }^{m - 8}$

$\sf \large\longmapsto {( \frac{3}{2}) }^{ - 6} = {( \frac{3}{2}) }^{m - 8}$

Same bases on both so it gets automatically cancelled

$\sf \longmapsto - 6 = m - 8$

$\sf \longmapsto \: m = - 6 + 8$

$\sf \longmapsto \bold{ m = 2}$

Check

$\sf \large \longmapsto {( \frac{2}{3}) }^{6} \times { (\frac{3}{2}) }^{10} = { (\frac{3}{2} )}^{m + 2}$

$\sf \large \longmapsto { (\frac{3}{2}) }^{ - 6} \times { (\frac{3}{2} )}^{10} = { (\frac{3}{2} )}^{4}$

$\sf \longmapsto \large \frac{3 \times 3 \times 3 \times 3}{2 \times 2 \times 2 \times 2} = \frac{81}{16}$

$\sf \longmapsto \large { (\frac{3}{2} )}^{m + 2} = {( \frac{3}{2} )}^{4} = \frac{81}{16}$