Question

describe and correct the error a student made in multiplying \\(\sqrt3{24}\\) and \\(\sqrt3{45}\\).
\\(\

$$\begin{align*}\\sqrt{24}\\cdot\\sqrt{45}&=\\sqrt{24\\cdot45}\\\\&=\\sqrt{2\\cdot2\\cdot2\\cdot3\\cdot3\\cdot3\\cdot5}\\\\&=2\\cdot3\\sqrt{2\\cdot3\\cdot5}\\\\&=6\\sqrt{30}\\end{align*}$$

\\) \\(\boldsymbol{\times}\\)

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.)
\\(\bigcirc\\) a. the error occurs in the third line of the students work because the student pulled out two factors of 2 and two factors of 3 from the radicand. the correct final answer is \\(\square\\).
\\(\bigcirc\\) b. the error occurs in the fourth line of the students work because the student multiplied the factors instead of adding them. the correct final answer is \\(\square\\).
\\(\bigcirc\\) c. the error occurs in the second line of the students work because the students prime factorization is incorrect. the correct final answer is \\(\square\\).
\\(\bigcirc\\) d. the error occurs in the first line of the students work because the student multiplied the radicands instead of adding them. the correct final answer is \\(\square\\).

Explanation:

Step1: Apply the product rule for cube - roots

The product rule for cube - roots is \(\sqrt[3]{a}\cdot\sqrt[3]{b}=\sqrt[3]{a\cdot b}\). So, \(\sqrt[3]{24}\cdot\sqrt[3]{45}=\sqrt[3]{24\times45}\).
First, find the prime factorization of \(24 = 2\times2\times2\times3\) and \(45=3\times3\times5\). Then \(24\times45=(2\times2\times2\times3)\times(3\times3\times5)=2^{3}\times3^{3}\times5\).

Step2: Simplify the cube - root

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

The student treated cube - roots as square - roots. The error occurs in the first line of the student's work because the rules for multiplying cube - roots are different from square - roots.

Answer:

A. 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 \(6\sqrt[3]{5}\)