Question
Given \( x>0 \), simplify \( \sqrt{100 x^{8}} \) completely.
Ask by Watson Schofield. in the United States
Feb 25,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The simplified form is \( 10x^{4} \).
Solution
To simplify √(100x⁸) when x > 0, first note that 100 is a perfect square, and so is x⁸, because 100 = 10² and x⁸ = (x⁴)². You can separate the square root as follows:
√(100x⁸) = √(100) × √(x⁸)
= 10 × x⁴
= 10x⁴
Thus, the simplified form is 10x⁴.
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Simplify this solution Extra Insights
To simplify \( \sqrt{100 x^{8}} \), we start by breaking it down. The square root of a product can be separated into the product of the square roots: \[ \sqrt{100 x^{8}} = \sqrt{100} \cdot \sqrt{x^{8}}. \] Now, simplifying each part, we have \( \sqrt{100} = 10 \) since \( 10 \times 10 = 100 \). For \( \sqrt{x^{8}} \), we can apply the property of exponents, which gives us \( \sqrt{x^{8}} = x^{4} \), because taking the square root of \( x^{8} \) means dividing the exponent by 2. Putting it all together, we find: \[ \sqrt{100 x^{8}} = 10 \cdot x^{4} = 10x^{4}. \] So, the fully simplified form is \( 10x^{4} \).
