Sure, let's carefully go through Jamal's simplification and identify the mistake.
We start with the original expression:
[tex]\[
\sqrt{75 x^5 y^8}
\][/tex]
First, we can break down the expression inside the square root:
[tex]\[
75 x^5 y^8 = 25 \cdot 3 \cdot x^4 \cdot x \cdot y^8
\][/tex]
So, the expression under square root becomes:
[tex]\[
\sqrt{25 \cdot 3 \cdot x^4 \cdot x \cdot y^8}
\][/tex]
We know that:
[tex]\[
\sqrt{a \cdot b \cdot c} = \sqrt{a} \cdot \sqrt{b} \cdot \sqrt{c}
\][/tex]
Therefore, we can separate each factor:
[tex]\[
\sqrt{25} \cdot \sqrt{3} \cdot \sqrt{x^4} \cdot \sqrt{x} \cdot \sqrt{y^8}
\][/tex]
We know the square roots of the individual terms are:
[tex]\[
\sqrt{25} = 5
\][/tex]
[tex]\[
\sqrt{3} = \sqrt{3}
\][/tex]
[tex]\[
\sqrt{x^4} = x^2 \quad (\text{since the square root of } x^4 \text{ is } x^2)
\][/tex]
[tex]\[
\sqrt{x} = \sqrt{x}
\][/tex]
[tex]\[
\sqrt{y^8} = y^4 \quad (\text{since the square root of } y^8 \text{ is } y^4)
\][/tex]
Combining all these, we get:
[tex]\[
5 \cdot x^2 \cdot y^4 \cdot \sqrt{3x}
\][/tex]
Therefore, the correct simplified form is:
[tex]\[
5 x^2 y^4 \sqrt{3x}
\][/tex]
Jamal, however, simplified the original expression to:
[tex]\[
5 x^2 y^2 \sqrt{3x}
\][/tex]
By comparing both expressions, we notice that Jamal wrote the square root of [tex]\( y^8 \)[/tex] as [tex]\( y^2 \)[/tex] instead of [tex]\( y^4 \)[/tex]. Hence, the correct description of the error Jamal made is:
He should have written the square root of [tex]\( y^8 \)[/tex] in the answer as [tex]\( y^4 \)[/tex], not [tex]\( y^2 \)[/tex].
