Question

describe and correct the error a student made in multiplying $sqrt3{24}$ and $sqrt3{45}$
$sqrt3{24} cdot sqrt3{45} = sqrt3{24 cdot 45}$
$= sqrt3{2 cdot 2 cdot 2 cdot 3 cdot 3 cdot 5}$
$= 2 cdot 3sqrt3{2 cdot 3 cdot 5}$
$= 6sqrt3{30}$

select the correct choice below, and fill in the answer box to complete your choice.
(type an exact answer, using radicals as needed. simplify your answer.)
a. the error occurs in the first line of the student’s work because the student multiplied the radicands instead of adding them. the correct final answer is

b. the error occurs in the fourth line of the student’s work because the student multiplied the factors instead of adding them. the correct final answer is

c. the error occurs in the third line of the student’s work because the student pulled out two factors of 2 and two factors of 3 from the radicand. the correct final answer is

d. the error occurs in the second line of the student’s work because the student’s prime factorization is incorrect. the correct final answer is

Explanation:

Step1: Recall the property of cube roots

For any real numbers \(a\) and \(b\), \(\sqrt[3]{a}\cdot\sqrt[3]{b}=\sqrt[3]{ab}\). So first, we multiply the radicands: \(\sqrt[3]{24}\cdot\sqrt[3]{45}=\sqrt[3]{24\times45}\)

Step2: Factorize the product

Calculate \(24\times45 = 1080\). Now factorize \(1080\) into prime factors: \(1080=2\times540=2\times2\times270=2\times2\times2\times135=2^3\times3\times45=2^3\times3\times3\times15=2^3\times3^3\times5\)

Step3: Simplify the cube root

Using the property \(\sqrt[3]{x^3}=x\), we have \(\sqrt[3]{2^3\times3^3\times5}=\sqrt[3]{2^3}\times\sqrt[3]{3^3}\times\sqrt[3]{5}=2\times3\times\sqrt[3]{5}=6\sqrt[3]{5}\)

Now, let's analyze the student's error. The student probably made a mistake in the number of factors taken out (for cube roots, we need three of a kind to take one out). Looking at the options, option C says the error is in the third line where the student pulled out two factors of 2 and two factors of 3 (which is wrong, we need three). So we use the correct factorization.

Answer:

C. The error occurs in the third line of the student’s work because the student pulled out two factors of 2 and two factors of 3 from the radicand. The correct final answer is \(6\sqrt[3]{5}\)