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Jamal simplified the expression [tex]\sqrt{75 x^5 y^8}[/tex] where [tex]x \geq 0[/tex] and [tex]y \geq 0[/tex].

[tex]\sqrt{75 x^5 y^8}=\sqrt{25 \cdot 3 \cdot x^4 \cdot x \cdot y^8}=5 x^2 y^2 \sqrt{3 x}[/tex]

Which describes the error Jamal made?

A. 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].

B. He should have written the square root of [tex]x^4[/tex] in the answer as [tex]x[/tex], not [tex]x^2[/tex].

C. He should have written the 5 inside of the radical in the answer.

D. He should have written the 3 outside of the radical in the answer.


Sagot :

To find the error Jamal made in simplifying the expression [tex]\(\sqrt{75 x^5 y^8}\)[/tex], let's break down the process step-by-step correctly.

1. Initial Expression:
[tex]\[ \sqrt{75 x^5 y^8} \][/tex]

2. Factor Inside the Square Root:
[tex]\[ 75 = 3 \times 25 \quad \text{so,} \quad 75 x^5 y^8 = 3 \times 25 \times x^4 \times x \times y^8 \][/tex]
Therefore:
[tex]\[ \sqrt{75 x^5 y^8} = \sqrt{3 \times 25 \times x^4 \times x \times y^8} \][/tex]

3. Separate the Radicals:
We can separate the square root of a product into the product of square roots:
[tex]\[ \sqrt{3 \times 25 \times x^4 \times x \times y^8} = \sqrt{3} \times \sqrt{25} \times \sqrt{x^4} \times \sqrt{x} \times \sqrt{y^8} \][/tex]

4. Simplify Each Term:
- [tex]\(\sqrt{25} = 5\)[/tex]
- [tex]\(\sqrt{x^4} = x^2\)[/tex], since [tex]\(x^4\)[/tex] is a perfect square.
- [tex]\(\sqrt{y^8} = y^4\)[/tex], since [tex]\(y^8\)[/tex] is a perfect square.

So we get:
[tex]\[ \sqrt{3} \times 5 \times x^2 \times \sqrt{x} \times y^4 \][/tex]

5. Combine Terms Outside the Radical:
[tex]\[ 5x^2y^4 \sqrt{3x} \][/tex]

However, Jamal simplified the expression as:
[tex]\[ 5 x^2 y^2 \sqrt{3 x} \][/tex]

Now, compare this to the correct simplification:
[tex]\[ 5 x^2 y^4 \sqrt{3 x} \][/tex]

Jamal's error is in the power of [tex]\(y\)[/tex]. He wrote the square root of [tex]\(y^8\)[/tex] as [tex]\(y^2\)[/tex] instead of [tex]\(y^4\)[/tex].

### Conclusion
The correct option 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]."