Question

find value of x and y​

Answer 1

$\mathsf{Given :\;\dfrac{\big[\sqrt{32}\big]^x}{(2)^{y + 1}} = 1}$

★  32 can be written as 2⁵

$\mathsf{\implies \dfrac{\big[\sqrt{2^5}\big]^x}{(2)^{y + 1}} = 1}$

$\mathsf{\implies \dfrac{\bigg[\big({2^5}\big)^{\dfrac{1}{2}}\bigg]^x}{(2)^{y + 1}} = 1}$

$\mathsf{\implies \dfrac{\bigg[\big({2}\big)^{\dfrac{5}{2}}\bigg]^x}{2^{y + 1}} = 1}$

$\mathsf{\implies \dfrac{\big({2}\big)^{\dfrac{5x}{2}}}{(2)^{y + 1}} = 1}$

$\bigstar\;\;\textsf{We know that : \boxed{\mathsf{\dfrac{a^m}{a^n} = a^{m - n}}}}$

$\mathsf{\implies \big[{2}\big]^{\dfrac{5x}{2} - [y + 1]}} = 1}$

$\mathsf{\implies \big[{2}\big]^{\dfrac{5x - 2[y + 1]}{2}}} = 1}$

★  1 can be expressed as 2⁰

$\mathsf{\implies \big[{2}\big]^{\dfrac{5x - 2[y + 1]}{2}} = \big[2\big]^0}$

$\mathsf{\implies {\dfrac{5x - 2[y + 1]}{2}} = 0}$

$\mathsf{\implies {5x - 2[y + 1]} = 0}$

$\mathsf{\implies {5x - 2y = 2}}$

$\mathsf{\implies x = \dfrac{2y + 2}{5}}$

$\mathsf{Given :\;\big[8\big]^{y} - \big[16\big]^{4 - \dfrac{x}{2}} = 0}$

$\mathsf{\implies \big[8\big]^{y} = \big[16\big]^{4 - \dfrac{x}{2}}}$

★  8 can be written as 2³

★  16 can be written as 2⁴

$\mathsf{\implies \big[2^{3}\big]^{y} = \big[2^{4}\big]^{4 - \dfrac{x}{2}}}$

$\mathsf{\implies \big[2\big]^{3y} = \big[2\big]^{4\bigg[4 - \dfrac{x}{2}\bigg]}}$

$\mathsf{\implies \big[2\big]^{3y} = \big[2\big]^{4\bigg[\dfrac{8 - x}{2}\bigg]}}$

$\mathsf{\implies \big[2\big]^{3y} = \big[2\big]^{2[{8 - x}]}}$

$\mathsf{\implies \big[2\big]^{3y} = \big[2\big]^{{16 - 2x}}}$

★  Bases are same on both sides, Exponents should be Equal

$\mathsf{\implies 3y = 16 - 2x}$

Substituting the Value of x in the above Equation, We get :

$\mathsf{\implies 3y = 16 - 2\bigg[\dfrac{2y + 2}{5}\bigg]}$

$\mathsf{\implies 3y = \dfrac{80 - 4y - 4}{5}}$

$\mathsf{\implies 15y = 80 - 4y - 4}}$

$\mathsf{\implies 15y + 4y = 80 - 4}}$

$\mathsf{\implies 19y = 76}}$

$\mathsf{\implies y = 4}}$

$\mathsf{Substituting\;the\;value\;of\;y\;in\;x = \dfrac{2y + 2}{5}, We\;get :}$

$\mathsf{\implies x = \dfrac{2(4) + 2}{5}}$

$\mathsf{\implies x = \dfrac{10}{5}}$

$\mathsf{\implies x = 2}$

Answers :

★  x = 2

★  y = 4