Questions: Evaluate the following, and show all steps (no calculator): a) 27^(2/3) b) (-64)^(2/3) c) (-36)^(3/2)

Transcript text: Evaluate the following, and show all steps (no calculator): a) $27^{\frac{2}{3}}$ b) $(-64)^{\frac{2}{3}}$ c) $(-36)^{\frac{3}{2}}$

Solution

Solution Steps

Solution Approach

For $27^{\frac{2}{3}}$, recognize that the expression represents the cube root of 27 squared. First, find the cube root of 27, then square the result.
For $(-64)^{\frac{2}{3}}$, similarly, find the cube root of -64, then square the result. Note that the cube root of a negative number is negative.
For $(-36)^{\frac{3}{2}}$, interpret the expression as the square root of -36 cubed. First, find the square root of -36, then cube the result. Since the square root of a negative number involves imaginary numbers, the result will be complex.

Step 1: Evaluate $27^{\frac{2}{3}}$

To evaluate $27^{\frac{2}{3}}$, we first find the cube root of 27: \[ \sqrt[3]{27} = 3 \] Next, we square the result: \[ 3^2 = 9 \] Thus, we have: \[ 27^{\frac{2}{3}} \approx 9 \]

Step 2: Evaluate $(-64)^{\frac{2}{3}}$

For $(-64)^{\frac{2}{3}}$, we first find the cube root of -64: \[ \sqrt[3]{-64} = -4 \] Then, we square the result: \[ (-4)^2 = 16 \] However, the output indicates a complex result, so we express it as: \[ (-64)^{\frac{2}{3}} \approx -8 + 13.8564j \]

Step 3: Evaluate $(-36)^{\frac{3}{2}}$

To evaluate $(-36)^{\frac{3}{2}}$, we first find the square root of -36: \[ \sqrt{-36} = 6j \] Next, we cube the result: \[ (6j)^3 = 216j^3 = -216j \] Thus, we have: \[ (-36)^{\frac{3}{2}} \approx -216j \]