Final answer:
Sarah incorrectly simplified the radical expression sqrt(48x^7 * y^2 * z); the correct answer is 4x^3 * y * sqrt(3z), not 2x^3 * y * sqrt(3z).
Step-by-step explanation:
The student is asking whether the simplification of the radical expression sqrt(48x^7 * y^2 * z) to 2x^3 * y * sqrt(3z) by Sarah is correct.
First, we should factor the number 48 to find perfect squares and then apply the properties of square roots. The prime factorization of 48 is 24×3, which can be further broken down into (22)2×3. Using the rule that xpxq = x(p+q), we can simplify the expression by taking the square root of each part separately.
For the x term, x7 can be written as (x3)2×x, taking the square root of (x3)2 gives us x3. For the y term, y2 is already a perfect square, so taking its square root gives us y. Finally, the square root of z remains as sqrt(z).
The correct simplification of sqrt(48x7 * y2 * z) is 4x3 * y * sqrt(3z), not 2x3 * y * sqrt(3z) as Sarah suggested. Therefore, Sarah's answer is incorrect.