determine the solution for the following equation: $(8x - 8)^{\frac{3}{2}}=64$ \n$x = 3$\n$x = 5$\n$x =…

Explanation:

Step1: Isolate the power - term

First, rewrite the equation ((8x - 8)^{\frac{3}{2}}=64). Let (y = 8x - 8), so we have (y^{\frac{3}{2}}=64).

Step2: Get rid of the fractional exponent

Raise both sides of the equation (y^{\frac{3}{2}} = 64) to the (\frac{2}{3}) power. According to the power - of - a - power rule ((a^{m})^{n}=a^{mn}), we get (y=(64)^{\frac{2}{3}}). Since (64 = 4^{3}), then ((64)^{\frac{2}{3}}=(4^{3})^{\frac{2}{3}}=4^{3\times\frac{2}{3}} = 4^{2}=16). So (y = 16).

Step3: Solve for (x)

Since (y = 8x - 8) and (y = 16), we set up the equation (8x-8 = 16). Add 8 to both sides: (8x=16 + 8=24). Divide both sides by 8: (x=\frac{24}{8}=3).

Answer:

A. (x = 3)