Question
Question 3 Assuming \( x \) and \( y \) are both positive, write the following expression in simplest radical form. 121 mins \[ x^{2} \sqrt{4 x^{4} y^{7}} \] Answer
Ask by Powell Davies. in the United States
Feb 25,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The expression simplifies to \( 2x^{4}y^{3}\sqrt{y} \).
Solution
We start with the expression:
x² √(4x⁴y⁷)
Since x and y are both positive, we can simplify the square root by factoring perfect squares.
1. Rewrite the square root:
√(4x⁴y⁷) = √(4) · √(x⁴) · √(y⁷)
= 2 · x² · √(y⁷)
(note that √(x⁴) = x² because x > 0)
2. Simplify √(y⁷):
y⁷ can be written as y⁶ · y = (y³)² · y, so
√(y⁷) = √((y³)² · y) = y³√y
3. Substitute back:
√(4x⁴y⁷) = 2x² (y³√y) = 2x²y³√y
4. Now, multiply this by the initial x²:
x² · (2x²y³√y) = 2x⁴y³√y
So, the expression in simplest radical form is 2x⁴y³√y.
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Simplify this solution Bonus Knowledge
First, we simplify the expression \( \sqrt{4 x^{4} y^{7}} \). We know that \( \sqrt{4} = 2 \) and \( \sqrt{x^{4}} = x^{2} \). For \( y^{7} \), we can express it as \( y^{6} \cdot y \) giving us \( \sqrt{y^{6}} = y^{3} \). So, we have: \[ \sqrt{4 x^{4} y^{7}} = 2x^{2}y^{3}\sqrt{y} \] Now, we can substitute this back into the original expression: \[ x^{2} \sqrt{4 x^{4} y^{7}} = x^{2} (2x^{2} y^{3} \sqrt{y}) = 2x^{4} y^{3} \sqrt{y} \] Thus, the expression in simplest radical form is: \[ 2x^{4} y^{3} \sqrt{y} \]
