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Simplify \(\frac {\sqrt{6}}{\sqrt{21}}\).
Before jumping to rationalize the denominator, try to simplify the expression by dividing the radicals first. In this case, both numerator and denominator have a common factor of three.
\(\frac{{\sqrt 6 }}{{\sqrt {21} }} = \frac{{\sqrt 2 }}{{\sqrt 7 }}\)
Then, rationalize the denominator.
Note that if you had not simplified first, you still would have arrived at the same solution, but you would have had to simplify the expression after rationalizing the denominator.
By simplifying first, you generally will end up working with smaller, more manageable numbers.
\(\frac{{\sqrt 6 }}{{\sqrt {21} }} = \frac{{\sqrt 2 }}{{\sqrt 7 }}\)
Then, rationalize the denominator.
\(\begin{align}
\frac{{\sqrt 2 }}{{\sqrt 7 }} &= \frac{{\sqrt 2 }}{{\sqrt 7 }}\cdot \frac{{\sqrt 7 }}{{\sqrt 7 }} \\
&= \frac{{\sqrt {14} }}{7} \\
\end{align}\)
\frac{{\sqrt 2 }}{{\sqrt 7 }} &= \frac{{\sqrt 2 }}{{\sqrt 7 }}\cdot \frac{{\sqrt 7 }}{{\sqrt 7 }} \\
&= \frac{{\sqrt {14} }}{7} \\
\end{align}\)
Note that if you had not simplified first, you still would have arrived at the same solution, but you would have had to simplify the expression after rationalizing the denominator.
By simplifying first, you generally will end up working with smaller, more manageable numbers.