The problem consists of two questions. Question 1 asks to evaluate five expressions involving exponents. Question 2 asks to solve five exponential equations for $x$.

1. Problem Description

The problem consists of two questions.

Question 1 asks to evaluate five expressions involving exponents.

Question 2 asks to solve five exponential equations for $x$.

2. Solution Steps

Question 1:

i. Evaluate $8^{\frac{2}{3}}$

$8^{\frac{2}{3}} = (8^{\frac{1}{3}})^2 = (2)^2 = 4$

ii. Evaluate $64^{\frac{1}{2}}$

$64^{\frac{1}{2}} = \sqrt{64} = 8$

iii. Evaluate $4^{-\frac{3}{2}}$

$4^{-\frac{3}{2}} = (4^{\frac{1}{2}})^{-3} = (2)^{-3} = \frac{1}{2^3} = \frac{1}{8}$

iv. Evaluate $(\frac{27}{8})^{-\frac{1}{3}}$

$(\frac{27}{8})^{-\frac{1}{3}} = (\frac{8}{27})^{\frac{1}{3}} = (\frac{2^3}{3^3})^{\frac{1}{3}} = \frac{2}{3}$

v. Evaluate $(\frac{16}{9})^{\frac{3}{2}}$

$(\frac{16}{9})^{\frac{3}{2}} = ((\frac{16}{9})^{\frac{1}{2}})^3 = (\frac{4}{3})^3 = \frac{4^3}{3^3} = \frac{64}{27}$

Question 2:

i. Solve $2^x = 4^{-1}$

$2^x = 4^{-1}$

$2^x = (2^2)^{-1}$

$2^x = 2^{-2}$

$x = -2$

ii. Solve $3 = 9^x$

$3 = 9^x$

$3 = (3^2)^x$

$3 = 3^{2x}$

$1 = 2x$

$x = \frac{1}{2}$

iii. Solve $4^x = 64^{-1}$

$4^x = 64^{-1}$

$4^x = (4^3)^{-1}$

$4^x = 4^{-3}$

$x = -3$

iv. Solve $3^x = 27$

$3^x = 27$

$3^x = 3^3$

$x = 3$

v. Solve $\frac{1}{5} = 25^x$

$\frac{1}{5} = 25^x$

$5^{-1} = (5^2)^x$

$5^{-1} = 5^{2x}$

$-1 = 2x$

$x = -\frac{1}{2}$

3. Final Answer

Question 1:

i. 4

ii. 8

iii. 1/8

iv. 2/3

v. 64/27

Question 2:

i. -2

ii. 1/2

iii. -3

iv. 3

v. -1/2