A. The square root of some fractions can be determined by finding the square root of the numerator and denominator separately.
Example:
\(\sqrt {\frac{{16}}{{25}}} = \frac{{\sqrt {16} }}{{\sqrt {25} }} = \frac{{\sqrt {{4^2}} }}{{\sqrt {{5^2}} }} = \frac{4}{5}\)
For any positive number x and y,
\(\sqrt {\frac{x}{y}} = \frac{{\sqrt x }}{{\sqrt y }}\)
In other words, the square root of a fraction is a fraction of square roots.B. Some fractions can be reduced to fractions with perfect squares as the numerator and denominator. Then, the square root of the simplified fraction can be determined as shown in the example below.
Example:
\(\sqrt {\frac{{18}}{{50}}} = \sqrt {\frac{9}{{25}}} = \frac{{\sqrt {{3^2}} }}{{\sqrt {{5^2}} }} = \frac{3}{5}\)
C. For a mixed number, change it to an improper fraction before finding the square root.
Example:
\(\sqrt {1\frac{{13}}{{36}}} = \sqrt {\frac{{49}}{{36}}} = \frac{{\sqrt {{7^2}} }}{{\sqrt {{6^2}} }} = \frac{7}{6} = 1\frac{1}{6}\)
Example:
Calculate the value of each of the following:
\(\begin{array}{l}{\rm{a)}}\,\,\sqrt {\frac{{25}}{{36}}} \\{\rm{b)}}\,\,\sqrt {\frac{{18}}{{32}}} \\{\rm{c)}}\,\,\sqrt {1\frac{{11}}{{25}}} \end{array}\)
Solution:
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