Question

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

Explanation:

Step1: Simplify \(2\sqrt{50a^{3}b^{2}}\)

$$\begin{align*} 2\sqrt{50a^{3}b^{2}}&=2\sqrt{25\times2\times a^{2}\times a\times b^{2}}\\ &=2\times5ab\sqrt{2a}\\ & = 10ab\sqrt{2a} \end{align*}$$

Step2: Simplify \(2\sqrt{200a^{3}}\)

$$\begin{align*} 2\sqrt{200a^{3}}&=2\sqrt{100\times2\times a^{2}\times a}\\ &=2\times10a\sqrt{2a}\\ &=20a\sqrt{2a} \end{align*}$$

Step3: Combine like - terms

Like - terms have the same radical part. Here, \(10ab\sqrt{2a}\) and \(20a\sqrt{2a}\) are not like - terms because the first term has a factor of \(b\) and the second does not. So, we cannot combine \(10ab\sqrt{2a}+20a\sqrt{2a}\) as \(30ab\sqrt{2a}\)

Answer:

The student incorrectly simplified \(10ab\sqrt{2a}+20a\sqrt{2a}\).