Answered

Jamal simplified the expression $\sqrt{75 x^5 y^8}$ where $x \geq 0$ and $y \geq 0$.

$\sqrt{75 x^5 y^8}=\sqrt{25 \cdot 3 \cdot x^4 \cdot x \cdot y^8}=5 x^2 y^2 \sqrt{3 x}$



Which describes the error Jamal made?


AHe should have written the square root of $y^8$ in the answer as $y^4$, not $y^2$.
BHe should have written the square root of $x^4$ in the answer as $x$, not $x^2$.
CHe should have written the 5 inside of the radical in the answer.
DHe should have written the 3 outside of the radical in the answer.

Verified Answer :

  • Analyze the given expression: \(\sqrt{75 x^5 y^8} = \sqrt{25 \cdot 3 \cdot x^4 \cdot x \cdot (y^4)^2}\).
  • Evaluate each term: \(\sqrt{25}=5\), \(\sqrt{x^4}=x^2\), and \(\sqrt{(y^4)^2}=y^4\).
  • Combine the results to get the correct simplification: \(5 x^2 y^4 \sqrt{3x}\).
  • Identify that Jamal incorrectly wrote \(y^2\) instead of \(y^4\), so the error is: \(\boxed{\text{He should have written the square root of } y^8 \text{ in the answer as } y^4, \text{ not } y^2.}\)

Explanation

  1. Analysis of the Expression

We are asked to identify the error in Jamal's simplification of the expression \(\sqrt{75 x^5 y^8}\), given \(x \geq 0\) and \(y \geq 0\). Jamal's simplification is:
\[\sqrt{75 x^5 y^8} = \sqrt{25 \cdot 3 \cdot x^4 \cdot x \cdot y^8} = 5 x^2 y^2 \sqrt{3 x}\]
We need to evaluate the square root of each factor correctly.

  1. Correct Simplification

To simplify \(\sqrt{75 x^5 y^8}\), we look at each term:

  • The coefficient: \(\sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}\).
  • The power of \(x\): \(\sqrt{x^5} = \sqrt{x^4 \times x} = \sqrt{(x^2)^2 \times x} = x^2\sqrt{x}\).
  • The power of \(y\): \(\sqrt{y^8} = \sqrt{(y^4)^2} = y^4\).

Combining these, we get:
\[\sqrt{75 x^5 y^8} = \sqrt{25 \times 3 \times x^4 \times x \times y^8} = \sqrt{25} \cdot \sqrt{3} \cdot \sqrt{x^4} \cdot \sqrt{x} \cdot \sqrt{y^8} = 5 \cdot \sqrt{3} \cdot x^2 \cdot \sqrt{x} \cdot y^4 = 5 x^2 y^4 \sqrt{3x}\]

  1. Identifying the Error

Comparing the correct result \(5 x^2 y^4 \sqrt{3x}\) with Jamal's result \(5 x^2 y^2 \sqrt{3x}\), we see that Jamal wrote \(y^2\) instead of \(y^4\). Specifically, Jamal took the square root of \(y^8\) to be \(y^2\), which is incorrect because \(\sqrt{y^8} = y^4\). Therefore, the error Jamal made is in the simplification of the term \(\sqrt{y^8}\).

Examples

Understanding how to correctly simplify radical expressions is fundamental in algebra. For example, if you are designing a square garden with an area given by an expression like \(75x^5y^8\), you would need to calculate the side length by taking the square root. Simplifying this correctly allows you to estimate the materials needed for the perimeter fencing. Knowing the rules for powers, such as \(\sqrt{y^{2n}} = y^n\), helps in real-world measurements and engineering applications where complex variables represent physical quantities.

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