C3 — because 27^(1/3) = 3 and then dividing by 3 gives... wait
D√(27²) = √729 ≈ 27
27^(2/3) = (27^(1/3))² = (∛27)² = 3² = 9. The rational exponent m/n means: take the nth root first (getting a smaller number), then raise to the m power. Taking the root first (∛27 = 3) keeps numbers manageable. Option A is the most common error — multiplying 27 by 2/3 as if it were ordinary multiplication, confusing the exponent with a coefficient. Rational exponents are exponents, not multipliers.
Question 2 Multiple Choice
A student wants to simplify √x · ∛x. Using rational exponents, what is the result?
Ax^(5/6), which equals ⁶√(x⁵)
Bx^(2/6) = x^(1/3), the smaller root dominates
Cx^(1/6) — you multiply the exponents when multiplying roots
D∜(x²), because you average the two root indices
Convert to rational exponents: √x = x^(1/2) and ∛x = x^(1/3). Multiplying: x^(1/2) · x^(1/3) = x^(1/2 + 1/3) = x^(3/6 + 2/6) = x^(5/6) = ⁶√(x⁵). This illustrates exactly why rational exponent notation is powerful — a product of roots with different indices is reduced to fraction addition. Option C reverses the product rule: you add exponents when multiplying, not multiply them (multiplying exponents is for the power rule, (x^a)^b = x^(ab)).
Question 3 True / False
x^(1/2) means x divided by 2.
TTrue
FFalse
Answer: False
This is the most common confusion with rational exponents. x^(1/2) means the square root of x — it is the number that, when squared, gives x. The '1/2' is the exponent (the power to which x is raised), not a coefficient or a divisor. x divided by 2 would be written x/2 or (1/2)x, not x^(1/2). For example, 9^(1/2) = 3 (the square root of 9), not 9/2 = 4.5.
Question 4 True / False
For any real number x, the expression x^(1/3) is defined as a real number.
TTrue
FFalse
Answer: True
Cube roots (and all odd roots) are defined for all real numbers, including negatives. For example, (−8)^(1/3) = −2, because (−2)³ = −8. This contrasts with even roots: x^(1/2) requires x ≥ 0 in the reals because squaring any real number gives a non-negative result, so no real number has a negative square. The even/odd distinction on the root index determines whether negative inputs are allowed.
Question 5 Short Answer
Explain why x^(1/2) must equal √x — not as an arbitrary definition, but as a logical consequence of the exponent rules you already know.
Think about your answer, then reveal below.
Model answer: The power rule says (x^a)^b = x^(ab). If we apply this to a = 1/2 and b = 2: (x^(1/2))² = x^(1/2 · 2) = x^1 = x. So x^(1/2) is a number that, when squared, gives x. That is exactly the definition of the square root. The exponent rules force x^(1/2) = √x — it is not a new definition pasted on, but what consistency with the existing rules requires. The same logic gives x^(1/n) = ⁿ√x for any positive integer n.
This is the conceptual heart of rational exponents: the notation is not arbitrary — it is the unique extension of the integer exponent rules that preserves consistency. Students who memorize 'x^(1/2) = √x' as a rule often don't see why it must be true. Understanding that the power rule forces this choice makes rational exponents feel inevitable rather than arbitrary, and it extends naturally to x^(m/n) without additional memorization.