the dimensions of a rectangle are $sqrt{50a^{3}b^{2}}$ and $sqrt{200a^{3}}$. a student found the perimeter…
the dimensions of a rectangle are $sqrt{50a^{3}b^{2}}$ and $sqrt{200a^{3}}$. a student found the perimeter as follows: $2sqrt{50a^{3}b^{2}}+2sqrt{200a^{3}} = 2cdot5absqrt{2a}+2cdot10asqrt{2a}=10absqrt{2a}+20asqrt{2a}=30absqrt{2a}$. what is the students error? the student should not have multiplied each expression by two. the student simplified $2sqrt{50a^{3}b^{2}}$ incorrectly. the student simplified $2sqrt{200a^{3}}$ incorrectly. the student incorrectly simplified $10absqrt{2a}+20asqrt{2a}$. done
Answer
Explanation:
Step1: Recall perimeter formula
The perimeter of a rectangle $P = 2l+2w$, where $l$ and $w$ are length and width. The student correctly started with $2\sqrt{50a^{3}b^{2}}+2\sqrt{200a^{3}}$.
Step2: Simplify $\sqrt{50a^{3}b^{2}}$
$\sqrt{50a^{3}b^{2}}=\sqrt{25\times2\times a^{2}\times a\times b^{2}} = 5ab\sqrt{2a}$, so $2\sqrt{50a^{3}b^{2}}=2\times5ab\sqrt{2a}=10ab\sqrt{2a}$.
Step3: Simplify $\sqrt{200a^{3}}$
$\sqrt{200a^{3}}=\sqrt{100\times2\times a^{2}\times a}=10a\sqrt{2a}$, so $2\sqrt{200a^{3}}=2\times 10a\sqrt{2a}=20a\sqrt{2a}$.
Step4: Combine like - terms
When combining $10ab\sqrt{2a}+20a\sqrt{2a}$, these are not like - terms because the first term has a factor of $b$ and the second does not. We cannot combine them as $30ab\sqrt{2a}$.
Answer:
The student incorrectly simplified $10ab\sqrt{2a}+20a\sqrt{2a}$.