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

Answer: 9

Question 2: Calculate the value of: 8^(3/2).

Answer: 16

Question 3: Express 125^(1/3) as a simple number.

Answer: 5

Question 1: Explain why (16)^(3/4) can be simplified to 2.

Answer: Because 16^(3/4) = (16^(1/4))^3, and 16^(1/4) = 2, so (2)^3=8 (correcting the previous answer to 8)

Question 2: Evaluate and explain any mistake in: (81)^(2/4).

Answer: 81^(2/4) = (81^(1/4))^2. 81^(1/4) is not 3; it is 3^(4/4)=3, so the correct value is 3^2=9.

Question 3: If (a)^(m/n) = 16, and n=2, what is the value of a^(m)?

Answer: a^(m) = 16^n = 16^2 = 256

Question 1: A scientist is working with a substance whose radioactive decay follows the model N = N_0 * (1/2)^(t/T), where T is the half-life. If T=4 years, what is the decay factor after 8 years? Express your answer as a fractional index.

Answer: The decay factor is (1/2)^(8/4) = (1/2)^2 = 1/4

Question 1: Prove that (81)^(1/4) * (16)^(1/4) = 2√2 by simplifying each term carefully.

Answer: 81^(1/4) = 3, 16^(1/4)=2, so product = 3*2=6, which equals 2√2 only if considering specific context; thus, verify step-by-step.

Question 2: If a^(m/n) = 16, and a=2, find the value of m/n.

Answer: m/n=4

Question 1: Student's calculation: (64)^(3/2) = 64^(1/2)^3 = 8^3=512. Is this correct? If not, correct the mistake.

Answer: The student misapplied the order of roots and powers. Correct calculation: 64^(3/2) = (64^(1/2))^3 = 8^3=512, which is correct; but the initial expression was miswritten. The mistake is in the notation, not the calculation.

Question 2: Evaluate (32)^(2/5). A student claims it equals 4. Is this correct? Explain why or why not.

Answer: Yes, because (32)^(2/5) = (32^(1/5))^2. 32^(1/5)=2, so the value is 2^2=4. The student's answer is correct.