Asked by k.
How does rewriting radicals in different forms help you communicate your answer?
Rikki says that the product cubed root of 3 x 5^3 x the cubed root of 5 is 3 x 5^2 , or 75. Explain rikki's error and write the correct product.
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GPT-5 mini
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Rewriting radicals as fractional exponents makes the algebra clear and avoids illegal “cancellations.”
Start with the product
∛3 × 5^3 × ∛5.
Write cube roots as 1/3 exponents:
3^(1/3) · 5^3 · 5^(1/3) = 3^(1/3) · 5^(3 + 1/3) = 3^(1/3) · 5^(10/3).
Combine the cube-root factors:
3^(1/3)·5^(1/3) = (3·5)^(1/3) = ∛15,
so the product = 5^3 · ∛15 = 125·∛15 ≈ 308.28.
Rikki’s error was treating the cube roots and 5^3 in a way that illegally “cancelled” or changed exponents (effectively replacing ∛3 and ∛5 by 3 and 5). You cannot do that; you must keep track of the fractional exponents and only combine like bases. The correct product is 125·∛15.
Start with the product
∛3 × 5^3 × ∛5.
Write cube roots as 1/3 exponents:
3^(1/3) · 5^3 · 5^(1/3) = 3^(1/3) · 5^(3 + 1/3) = 3^(1/3) · 5^(10/3).
Combine the cube-root factors:
3^(1/3)·5^(1/3) = (3·5)^(1/3) = ∛15,
so the product = 5^3 · ∛15 = 125·∛15 ≈ 308.28.
Rikki’s error was treating the cube roots and 5^3 in a way that illegally “cancelled” or changed exponents (effectively replacing ∛3 and ∛5 by 3 and 5). You cannot do that; you must keep track of the fractional exponents and only combine like bases. The correct product is 125·∛15.
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