Question
4)) \( \left.{ }^{\frac{1}{x}}\right]_{j} \) Simplify. Assume \( b \) is greater than or equal to zero. \( \sqrt{75 b^{3}} \)
Ask by Wagner Romero. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( \sqrt{75b^3} = 5b\sqrt{3b} \)
Solution
Calculate or simplify the expression \( \sqrt{75b^3} \).
Simplify the expression by following steps:
- step0: Solution:
\(\sqrt{75b^{3}}\)
- step1: Write as a product:
\(\sqrt{25\times 3b^{3}}\)
- step2: Write in exponential form:
\(\sqrt{5^{2}\times 3b^{3}}\)
- step3: Rewrite the exponent:
\(\sqrt{5^{2}\times 3b^{2+1}}\)
- step4: Expand the expression:
\(\sqrt{5^{2}\times 3b^{2}\times b}\)
- step5: Reorder the terms:
\(\sqrt{5^{2}b^{2}\times 3b}\)
- step6: Use the properties of radicals:
\(\sqrt{5^{2}b^{2}}\times \sqrt{3b}\)
- step7: Simplify the root:
\(5b\sqrt{3b}\)
The simplified form of \( \sqrt{75b^3} \) is \( 5b\sqrt{3b} \).
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
To simplify \( \sqrt{75 b^{3}} \), we start by breaking it down into parts we can more easily work with. We can express 75 as \( 25 \times 3 \), so: \[ \sqrt{75 b^{3}} = \sqrt{25 \times 3 \times b^{3}} \] Next, we can use the property \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \): \[ = \sqrt{25} \cdot \sqrt{3} \cdot \sqrt{b^{3}} \] We know that \( \sqrt{25} = 5 \), and \( \sqrt{b^{3}} = b^{\frac{3}{2}} = b^{1} \cdot \sqrt{b} \). Thus, we can combine everything: \[ = 5 \sqrt{3} b^{1} \sqrt{b} = 5b\sqrt{3b} \] So, our fully simplified expression is: \[ 5b\sqrt{3b} \]
