Question

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

$$\begin{align*}\\sqrt3{24}\\cdot\\sqrt3{45}&=\\sqrt3{24\\cdot45}\\\\&=\\sqrt3{2\\cdot2\\cdot2\\cdot3\\cdot3\\cdot3\\cdot5}\\\\&=2\\cdot3\\sqrt3{2\\cdot3\\cdot5}\\\\&=6\\sqrt3{30}\\end{align*}$$

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

Explanation:

Step1: Analyze the multiplication of radicals

The formula for multiplying cube roots is $\sqrt[3]{a} \cdot \sqrt[3]{b}=\sqrt[3]{a \cdot b}$. The first step $\sqrt[3]{24} \cdot \sqrt[3]{45}=\sqrt[3]{24 \cdot 45}$ is correct.

Step2: Factorize the product

Calculate $24\times45 = 1080$. Now factorize $1080$: $1080=2\times2\times2\times3\times3\times3\times5$. This step is correct.

Step3: Simplify the cube root

For a cube root $\sqrt[3]{x^3\cdot y}=\ x\sqrt[3]{y}$. Here, we have three $2$s and three $3$s, so we can take out $2$ and $3$ from the cube root. So $\sqrt[3]{2\cdot2\cdot2\cdot3\cdot3\cdot3\cdot5}=2\times3\times\sqrt[3]{5}$? Wait, no, wait. Wait, the original student's step: $= 2\cdot3\sqrt[3]{2\cdot3\cdot5}$ is wrong. Wait, let's recalculate. Wait, $24\times45 = 1080$. Factorizing $1080$: $24 = 2^3\times3$, $45=3^2\times5$, so $24\times45=2^3\times3\times3^2\times5=2^3\times3^3\times5$. Oh! Wait, the student factorized $24\times45$ as $2\times2\times2\times3\times3\times3\times5$, which is $2^3\times3^3\times5$, that's correct. Then, when simplifying the cube root, $\sqrt[3]{2^3\times3^3\times5}=2\times3\times\sqrt[3]{5}$, because $\sqrt[3]{2^3}=2$, $\sqrt[3]{3^3}=3$, and the remaining factor is $5$. But the student wrote $\sqrt[3]{2\times3\times5}$, which is the error. The student incorrectly included extra factors inside the cube root. The correct simplification should be taking out the perfect cubes (the $2^3$ and $3^3$) and leaving only the non - perfect cube factor (which is $5$) inside the cube root.

Step4: Correct the simplification

So $\sqrt[3]{2^3\times3^3\times5}=2\times3\times\sqrt[3]{5}=6\sqrt[3]{5}$? Wait, no, wait, $24\times45 = 1080$, $2^3\times3^3\times5=8\times27\times5 = 1080$, yes. So $\sqrt[3]{2^3\times3^3\times5}=2\times3\times\sqrt[3]{5}=6\sqrt[3]{5}$. But the student had $\sqrt[3]{2\times3\times5}$ inside, which is wrong. The student multiplied the remaining factors incorrectly. The correct simplification is:

$\sqrt[3]{24}\cdot\sqrt[3]{45}=\sqrt[3]{24\times45}=\sqrt[3]{1080}=\sqrt[3]{2^3\times3^3\times5}=2\times3\times\sqrt[3]{5}=6\sqrt[3]{5}$

Wait, but let's check the student's steps again. The student's step: $= 2\cdot3\sqrt[3]{2\cdot3\cdot5}$ is wrong because after taking out $2^3$ and $3^3$, the remaining factor is $5$, not $2\times3\times5$. The student made a mistake in the factors left inside the cube root. The correct calculation is:

1. Start with $\sqrt[3]{24}\cdot\sqrt[3]{45}$
2. Apply the cube - root multiplication rule: $\sqrt[3]{24\times45}=\sqrt[3]{1080}$
3. Factorize $1080$ as $2^3\times3^3\times5$
4. Simplify the cube root: $\sqrt[3]{2^3\times3^3\times5}=2\times3\times\sqrt[3]{5}=6\sqrt[3]{5}$

The student's error was in the step where they simplified the cube root. They incorrectly included $2$ and $3$ inside the cube root after taking out the perfect cubes. The correct simplification should leave only the non - perfect cube factor (which is $5$) inside the cube root.

Answer:

The error is in the step where the student simplified $\sqrt[3]{2\cdot2\cdot2\cdot3\cdot3\cdot3\cdot5}$. The student incorrectly wrote $\sqrt[3]{2\cdot3\cdot5}$ inside the cube root after taking out $2$ and $3$. The correct simplification is $\sqrt[3]{2^3\times3^3\times5}=2\times3\times\sqrt[3]{5}=6\sqrt[3]{5}$. So the correct product of $\sqrt[3]{24}$ and $\sqrt[3]{45}$ is $6\sqrt[3]{5}$.