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?
A. He should have written the square root of $y^8$ in the answer as $y^4$, not $y^2$.
B. He should have written the square root of $x^4$ in the answer as $x$, not $x^2$.
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.
Answer (2)
Jamal simplified the expression 75 x 5 y 8 incorrectly by stating y 8 as y 2 instead of the correct y 4 . Therefore, his error was that he should have written the square root of y 8 in the answer as y 4 , not y 2 . The correct multiple choice option is A.
;
Break down the expression under the square root into its factors.
Take the square root of each factor.
Combine the simplified factors to get the correct expression: 5 x 2 y 4 3 x .
Identify that Jamal incorrectly simplified y 8 as y 2 instead of y 4 , so the correct answer is: He should have written the square root of y 8 in the answer as y 4 , not y 2 .
The final answer 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
Analyze the Problem We are given Jamal's simplification of the expression 75 x 5 y 8 as 5 x 2 y 2 3 x and need to identify his mistake. Let's correctly simplify the expression step by step.
Simplify the Expression First, we can break down the expression under the square root into its factors: 75 x 5 y 8 = 25 ⋅ 3 ⋅ x 4 ⋅ x ⋅ y 8 Now, we take the square root of each factor: 25 = 5 3 = 3 x 4 = x 2 x = x y 8 = y 4 So, the correct simplification is: 5 ⋅ 3 ⋅ x 2 ⋅ x ⋅ y 4 = 5 x 2 y 4 3 x
Identify the Error Comparing this with Jamal's simplification, 5 x 2 y 2 3 x , we see that Jamal made an error in taking the square root of y 8 . He wrote y 2 instead of y 4 .
State the Error Therefore, the error Jamal made was that he should have written the square root of y 8 in the answer as y 4 , not y 2 .
Examples
Square roots are used in many real-world applications, such as calculating distances using the Pythagorean theorem. For example, if you are building a rectangular garden and know the lengths of two sides, you can use the Pythagorean theorem to find the length of the diagonal. Simplifying square roots, like in this problem, helps in obtaining precise measurements and efficient designs in construction and landscaping.
