Question

the dimensions of a rectangle are \\(\sqrt{50a^3b^2}\\) and \\(\sqrt{200a^3}\\).
a student found the perimeter as follows:
student’s work
\\(2\sqrt{50a^3b^2} + 2\sqrt{200a^3} = 2 \cdot 5ab\sqrt{2a} + 2 \cdot 10a\sqrt{2a}\\)
\\(= 10ab\sqrt{2a} + 20a\sqrt{2a}\\)
\\(= 30ab\sqrt{2a}\\)
what is the student’s error?
the student incorrectly simplified \\(10ab\sqrt{2a} + 20a\sqrt{2a}\\).
the student simplified \\(2\sqrt{50a^3b^2}\\) incorrectly.
the student simplified \\(2\sqrt{200a^3}\\) incorrectly.
the student should not have multiplied each expression by two.

Explanation:

Step1: Analyze the perimeter formula

The perimeter of a rectangle is \( P = 2l + 2w \), where \( l \) and \( w \) are the length and width. Here, \( l=\sqrt{50a^{3}b^{2}} \) and \( w = \sqrt{200a^{3}} \), so the formula \( 2\sqrt{50a^{3}b^{2}}+2\sqrt{200a^{3}} \) is correct.

Step2: Simplify \( 2\sqrt{50a^{3}b^{2}} \)

Simplify \( \sqrt{50a^{3}b^{2}} \):

$$
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$$

Then \( 2\sqrt{50a^{3}b^{2}}=2\times5ab\sqrt{2a} = 10ab\sqrt{2a} \), which is correct.

Step3: Simplify \( 2\sqrt{200a^{3}} \)

Simplify \( \sqrt{200a^{3}} \):

$$
LATEXBLOCK1
$$

Then \( 2\sqrt{200a^{3}}=2\times10a\sqrt{2a}=20a\sqrt{2a} \), which is correct.

Step4: Combine the terms

The student has \( 10ab\sqrt{2a}+20a\sqrt{2a} \). These two terms are not like terms (one has a factor of \( b \) and the other does not), so they cannot be combined by adding the coefficients and keeping the same radical part. The student incorrectly simplified \( 10ab\sqrt{2a}+20a\sqrt{2a} \) to \( 30ab\sqrt{2a} \).

Answer:

The student incorrectly simplified \( 10ab\sqrt{2a}+20a\sqrt{2a} \). (The first option in the list of what the student's error is)