Question
54. \( \frac{4 \sqrt{5}}{3 \sqrt{8}} \)
Ask by Perkins Mccoy. in the United States
Mar 16,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The simplified form is \( \frac{\sqrt{10}}{3} \).
Solution
Calculate or simplify the expression \( \frac{4 \sqrt{5}}{3 \sqrt{8}} \).
Calculate the value by following steps:
- step0: Calculate:
\(\frac{4\sqrt{5}}{3\sqrt{8}}\)
- step1: Simplify the root:
\(\frac{4\sqrt{5}}{3\times 2\sqrt{2}}\)
- step2: Multiply the terms:
\(\frac{4\sqrt{5}}{6\sqrt{2}}\)
- step3: Reduce the fraction:
\(\frac{2\sqrt{5}}{3\sqrt{2}}\)
- step4: Multiply by the Conjugate:
\(\frac{2\sqrt{5}\times \sqrt{2}}{3\sqrt{2}\times \sqrt{2}}\)
- step5: Multiply the numbers:
\(\frac{2\sqrt{10}}{3\sqrt{2}\times \sqrt{2}}\)
- step6: Multiply the numbers:
\(\frac{2\sqrt{10}}{6}\)
- step7: Reduce the fraction:
\(\frac{\sqrt{10}}{3}\)
The simplified form of \( \frac{4 \sqrt{5}}{3 \sqrt{8}} \) is \( \frac{\sqrt{10}}{3} \).
Answered by UpStudy AI and reviewed by a Professional Tutor
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Simplify this solution Beyond the Answer
To simplify \( \frac{4 \sqrt{5}}{3 \sqrt{8}} \), you first need to simplify the denominator. Since \( \sqrt{8} = \sqrt{4 \cdot 2} = 2 \sqrt{2} \), we can rewrite the expression as: \[ \frac{4 \sqrt{5}}{3 \cdot 2 \sqrt{2}} = \frac{4 \sqrt{5}}{6 \sqrt{2}} = \frac{2 \sqrt{5}}{3 \sqrt{2}} \] Next, we can rationalize the denominator by multiplying the numerator and denominator by \( \sqrt{2} \): \[ \frac{2 \sqrt{5} \cdot \sqrt{2}}{3 \sqrt{2} \cdot \sqrt{2}} = \frac{2 \sqrt{10}}{3 \cdot 2} = \frac{\sqrt{10}}{3} \] So, \( \frac{4 \sqrt{5}}{3 \sqrt{8}} = \frac{\sqrt{10}}{3} \).
