Answer :
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]."
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]."
