Question

If 3^x+y = 27 and 27³ = 3^x-y, then find value of x² - y².​

Answer 1

given

${3}^{x + y} = 27 \: \: \: and \: \: \: {27}^{3} = {3}^{x - y}$

Now

${3}^{x + y} = 27 \\ \\ {3}^{x + y} = {3}^{3} \\ \\ x + y = 3 - - - (1)$

And

$\implies {27}^{3} = {3}^{x - y} \\ \\ \implies ( {3}^{3} ) ^{3} = {3}^{x - y} \\ \\ \implies {3}^{9} = {3}^{x - y} \\ \\ x - y = 9 - - - (2)$

Now multiply equation (1) and (2)

$(x + y)(x - y) = 3 \times 9 \\ \\ {x}^{2} - {y}^{2} = 27$