Question

(24/5)^-4÷ (12/15)^6 × (6/4)^5​

Answer 1

Given to solve :-

$\dfrac{\bigg(\dfrac{24}{5}\bigg)^{-4} }{\bigg(\dfrac{12}{15}\bigg)^{6} } \times \bigg(\dfrac{6}{4} \bigg)^5$

SOLUTION:-

As we know from exponent and laws,

$\dfrac{a^m}{b^m} = \bigg(\dfrac{a}{b} \bigg)^m$and

$\bigg(\dfrac{a}{b} \bigg)^{-n}=\bigg(\dfrac{b}{a} \bigg)^n$

So,

$\bigg(\dfrac{24}{5} \bigg)^{-4}=\bigg(\dfrac{5}{24} \bigg)^4$

$\bigg(\dfrac{12}{15} \bigg)^6=\dfrac{12^6}{15^6}$

$\bigg(\dfrac{6}{4} \bigg)^5=\dfrac{6^5}{4^5}$

Substituting the values we get

$\dfrac{\bigg(\dfrac{5^4}{24^4} \bigg)}{\bigg(\dfrac{12^6}{15^6}\bigg) }\times \bigg(\dfrac{6^5}{4^5}\bigg)$

$\dfrac{5^4}{24^4}\times\dfrac{15^6}{12^6}\times \dfrac{6^5}{4^5}$

$24^4 = (2\times12)^4= 2^4 \times 12^4$

$15^6= (3\times5)^6= 3^6 \times 5^6$

$12^6=(3\times4)^6= 3^6\times4^6$

$\dfrac{5^4}{ 2^4 \times 12^4}\times \dfrac{3^6\times5^6}{12^6} \times\dfrac{6^5}{4^5}$

$\dfrac{5^{10}\times3^6\times2^5\times3^5}{2^4\times12^{10}\times2^{10}}$

$\dfrac{5^{10}\times3^{11}\times2^5}{12^{10}\times2^{14}}$

$\dfrac{5^{10}\times3^{11}\times2^5}{3^{10}\times4^{10}\times2^{14}}$

$\dfrac{5^{10}\times3\times2^5}{4^{10}\times2^{14}}$

$\dfrac{5^{10}\times2^5\times3}{(2\times2)^{10}\times2^{14}}$

$\dfrac{5^{10}\times2^5\times3}{2^{20}\times2^{14}}$

$\dfrac{5^{10}\times3}{2^{29}}$

Required answer is 5¹⁰×3/2²⁹

Refer attachment for better explanation

Used formulae :-

$\dfrac{a^m}{b^m} = \bigg(\dfrac{a}{b} \bigg)^m$and

$\bigg(\dfrac{a}{b} \bigg)^{-n}=\bigg(\dfrac{b}{a} \bigg)^n$

${(a\times b)^n = a^n \times b^n}$