Question

if 3^(x+y) = 81 and 81^(x-y)= 3⁸ , then the find values of X and y respectively.

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Answer 1

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Answer 2

Explanation:

$\mathsf{Given : 3^{x + y} = 81}$

$\mathsf{\bigstar\;\;81\;can\;be\;written\;as : 3^4}$

$\mathsf{\implies 3^{x + y} = 3^4}$

$\bigstar\;\;\textsf{When Bases are same on both sides, Exponents should be equal}$

$\mathsf{\implies x + y = 4\;-----\;(1)}$

$\mathsf{Given : 81^{x - y} = 3^{8}}$

$\mathsf{\implies (3^4)^{x - y} = 3^{8}}$

$\mathsf{\implies 3^{4(x - y)} = 3^{8}}$

$\mathsf{\implies 4(x - y) = 8}$

$\mathsf{\implies (x - y) = \dfrac{8}{4}}$

$\mathsf{\implies x - y = 2\;-----\;(2)}$

$\textsf{Adding Equations (1) and (2), We get :}$

$\mathsf{\implies (x + y) + (x - y) = 4 + 2}$

$\mathsf{\implies x + x + y - y = 6}$

$\mathsf{\implies 2x = 6}$

$\mathsf{\implies x = 3}$

$\textsf{Substituting x = 3 in Equation (1), We get :}$

$\mathsf{\implies 3 + y = 4}$

$\mathsf{\implies y = 4 - 3}$

$\mathsf{\implies y = 1}$

Answers :

★  x = 3

★  y = 1