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Answer :
We start by converting the mixed number
[tex]$$3\frac{1}{25}$$[/tex]
into an improper fraction. The whole number part is 3 and the fractional part is [tex]$\frac{1}{25}$[/tex]. This conversion is done by multiplying the whole number by the denominator and then adding the numerator:
[tex]$$
3\frac{1}{25} = \frac{3 \times 25 + 1}{25} = \frac{75 + 1}{25} = \frac{76}{25}.
$$[/tex]
Next, we want to find the square root of this fraction. Recall that the square root of a fraction is the fraction of the square roots:
[tex]$$
\sqrt{\frac{76}{25}} = \frac{\sqrt{76}}{\sqrt{25}}.
$$[/tex]
Since [tex]$\sqrt{25} = 5$[/tex], the expression becomes:
[tex]$$
\frac{\sqrt{76}}{5}.
$$[/tex]
Notice that 76 can be factored as [tex]$4 \times 19$[/tex], and since 4 is a perfect square we have:
[tex]$$
\sqrt{76} = \sqrt{4 \times 19} = \sqrt{4} \cdot \sqrt{19} = 2\sqrt{19}.
$$[/tex]
Substituting this back into our square root expression yields:
[tex]$$
\sqrt{\frac{76}{25}} = \frac{2\sqrt{19}}{5}.
$$[/tex]
Finally, as a numerical approximation:
[tex]$$
\sqrt{\frac{76}{25}} \approx 1.7436.
$$[/tex]
Thus, the square root of [tex]$3\frac{1}{25}$[/tex] is given by
[tex]$$
\boxed{\frac{2\sqrt{19}}{5} \approx 1.7436.}
$$[/tex]
[tex]$$3\frac{1}{25}$$[/tex]
into an improper fraction. The whole number part is 3 and the fractional part is [tex]$\frac{1}{25}$[/tex]. This conversion is done by multiplying the whole number by the denominator and then adding the numerator:
[tex]$$
3\frac{1}{25} = \frac{3 \times 25 + 1}{25} = \frac{75 + 1}{25} = \frac{76}{25}.
$$[/tex]
Next, we want to find the square root of this fraction. Recall that the square root of a fraction is the fraction of the square roots:
[tex]$$
\sqrt{\frac{76}{25}} = \frac{\sqrt{76}}{\sqrt{25}}.
$$[/tex]
Since [tex]$\sqrt{25} = 5$[/tex], the expression becomes:
[tex]$$
\frac{\sqrt{76}}{5}.
$$[/tex]
Notice that 76 can be factored as [tex]$4 \times 19$[/tex], and since 4 is a perfect square we have:
[tex]$$
\sqrt{76} = \sqrt{4 \times 19} = \sqrt{4} \cdot \sqrt{19} = 2\sqrt{19}.
$$[/tex]
Substituting this back into our square root expression yields:
[tex]$$
\sqrt{\frac{76}{25}} = \frac{2\sqrt{19}}{5}.
$$[/tex]
Finally, as a numerical approximation:
[tex]$$
\sqrt{\frac{76}{25}} \approx 1.7436.
$$[/tex]
Thus, the square root of [tex]$3\frac{1}{25}$[/tex] is given by
[tex]$$
\boxed{\frac{2\sqrt{19}}{5} \approx 1.7436.}
$$[/tex]
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