Question

find the value of m​

Answer 1

Answer :-

Value of m is (D) 1.

Explanation :-

$\mathsf{( m^3 + 2^3 + 3^3) ^{ - \dfrac{5}{2} } = (6)^{ - 5} }$

$\mathsf{ \implies ( m^3 + 8 + 27) ^{ - \dfrac{5}{2} } = (6)^{ - 5} }$

$\mathsf{ \implies ( m^3 + 35) ^{ - \dfrac{5}{2} } = (6)^{ - 5} }$

Squaring on both sides

$\mathsf{ \implies ( m^3 + 35) ^{ - 5} = (6^{ - 5} })^2$

$\mathsf{ \implies ( m^3 + 35) ^{ - 5} = 6^{ - 5(2)} }$

$\boxed{ \bf \because ({a}^{m}) ^n = {a}^{mn} }$

$\mathsf{ \implies ( m^3 + 35) ^{ - 5} = 6^{ -10}}$

Raising to power 1/5 on both sides

$\mathsf{ \implies \{( m^3 + 35) ^{ - 5} \}^{ \dfrac{1}{5} } = (6^{ -10} )^{ \dfrac{1}{5} } }$

$\mathsf{ \implies ( m^3 + 35) ^{ - 1} = 6^{ -2} }$

$\mathsf{ \implies \dfrac{1}{( m^3 + 35) ^1 }= \dfrac{1}{6^2} }$

$\boxed{ \bf \because {a}^{ - m} = \dfrac{1}{ {a}^{m} } }$

$\mathsf{ \implies \dfrac{1}{ m^3 + 35 }= \dfrac{1}{6^2} }$

$\mathsf{ \implies m^3 + 35 =6^2 }$

$\mathsf{ \implies m^3 + 35 =36 }$

$\mathsf{ \implies m^3 =36 - 35 }$

$\mathsf{ \implies m^3 =1}$

$\mathsf{ \implies m = \sqrt[3]{1} }$

$\mathsf{ \implies m =1 }$

∴ the value of m is (D) 1.