If 0 x and 81^ ^2 x +81^ ^2 x =30 Then x takes the value

Step-by-step solution

Given the equation 81^ ^2 x + 81^ ^2 x = 30 with domain 0 x . Using the identity ^2 x + ^2 x = 1 , substitute ^2 x = 1 - ^2 x to rewrite the equation as 81^ ^2 x + 81^ 1 - ^2 x = 30 . Apply exponent rules to express it as y + 81 y = 30 , where y = 81^ ^2 x and y > 0 . Multiply through by y to form the quadratic equation y^2 - 30y + 81 = 0 , which factors as (y - 3)(y - 27) = 0 , yielding y = 3 or y = 27 . For y = 3 , 81^ ^2 x = 3 . Since 81 = 3^4 , this becomes 3^ 4 ^2 x = 3^1 , so 4 ^2 x = 1 and ^2 x = 1 4 . In th