solve for x in the following equation. (16)^{x - 2}=(64)^{5x} a. -\\frac{5}{13} b. -\\frac{4}{13} c…

solve for x in the following equation. (16)^{x - 2}=(64)^{5x} a. -\\frac{5}{13} b. -\\frac{4}{13} c. -\\frac{2}{13} d. -\\frac{3}{4}

solve for x in the following equation. (16)^{x - 2}=(64)^{5x} a. -\\frac{5}{13} b. -\\frac{4}{13} c. -\\frac{2}{13} d. -\\frac{3}{4}

Answer

Explanation:

Step1: Express 16 and 64 as powers of 2

Since (16 = 2^4) and (64=2^6), the equation ((16)^{x - 2}=(64)^{5x}) becomes ((2^4)^{x - 2}=(2^6)^{5x}). Using the power - of - a - power rule ((a^m)^n=a^{mn}), we get (2^{4(x - 2)}=2^{6\times5x}).

Step2: Set the exponents equal

Since (a^m=a^n) implies (m = n) for (a>0,a\neq1), we set (4(x - 2)=6\times5x). Expand the left - hand side: (4x-8 = 30x).

Step3: Solve for (x)

Subtract (4x) from both sides: (-8=30x - 4x). Simplify the right - hand side: (-8 = 26x). Divide both sides by 26: (x=\frac{-8}{26}=-\frac{4}{13}).

Answer:

B. (-\frac{4}{13})