We need to evaluate the square root of the mixed number $75 \frac{46}{49}$.

ArithmeticSquare RootFractionsMixed NumbersArithmetic Operations
2025/3/30

1. Problem Description

We need to evaluate the square root of the mixed number 75464975 \frac{46}{49}.

2. Solution Steps

First, we convert the mixed number to an improper fraction.
754649=75×49+4649=3675+4649=37214975 \frac{46}{49} = \frac{75 \times 49 + 46}{49} = \frac{3675 + 46}{49} = \frac{3721}{49}
Then, we find the square root of the fraction.
754649=372149\sqrt{75 \frac{46}{49}} = \sqrt{\frac{3721}{49}}
We know that ab=ab\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}, so we can rewrite the expression as:
372149=372149\sqrt{\frac{3721}{49}} = \frac{\sqrt{3721}}{\sqrt{49}}
We know that 49=7\sqrt{49} = 7.
We also know that 3721=61\sqrt{3721} = 61, since 61×61=372161 \times 61 = 3721.
Therefore,
372149=617\frac{\sqrt{3721}}{\sqrt{49}} = \frac{61}{7}
Now, we convert the improper fraction to a mixed number.
617=857\frac{61}{7} = 8 \frac{5}{7}
So, the square root of 75464975 \frac{46}{49} is 8578 \frac{5}{7}.

3. Final Answer

8578 \frac{5}{7}

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