Question 1: Calculate the value of 8^(1/3).

Answer: 2

Question 2: Simplify the expression: (27)^(1/3).

Answer: 3

Question 3: Evaluate: (16)^(1/4).

Answer: 2

Question 4: Calculate the value of 81^(1/4).

Answer: 3

Question 1: Express 125^(1/3) as a simplified radical and as a fractional power. Explain your steps.

Answer: Radical: ³√125, Fractional power: 125^(1/3)

Question 2: A square root is equivalent to which fractional index? If √a = a^(1/2), what is the value of (a)^(1/2)?

Answer: a^(1/2)

Question 3: If a = 16, what is the value of a^(1/4), and how does it relate to the radical form?

Answer: 2; ⁴√16=2

Question 4: Identify and correct the mistake in this statement: (64)^(1/3) = 4.

Answer: Correct value is ⁶√64=4, but (64)^(1/3)=Cubed root of 64=4, so the statement is correct. Mistake: misinterpreting the root as a power.

Question 1: A tank is filled to a volume proportional to the cube of a certain measurement. If the measurement is 4 units, what is the measurement if the volume is doubled? Use fractional indices to justify.

Answer: Measurement = (Volume)^(1/3); if volume doubles, measurement = 2^(1/3)×4.

Question 1: Determine the value of (x^3)^(1/6) and simplify your answer.

Answer: x^(3×1/6)=x^(1/2)

Question 2: Explain why (a^2)^(1/2) = a, and discuss any assumptions made about a.

Answer: Since (a^2)^(1/2)=a^(2×1/2)=a^1=a, assuming a≥0 for real roots.

Question 1: A student claims that (16)^(1/2) = 4. Is this correct? Explain your reasoning.

Answer: Yes, because ⁴√16=4 and (16)^(1/2)=√16=4.

Question 2: Identify the mistake: (81)^(1/4) = 3. Is this correct? Why or why not?

Answer: Incorrect; (81)^(1/4) is the 4th root of 81, which is ³√81≈4.32, not 3.

Question 3: A common mistake is treating a^(1/n) as a×n. Why is this wrong?

Answer: Because a^(1/n) equals the n-th root of a, not a multiplied by n.