It is really easy to find the square root of a fraction if the numerator and denominator are perfect squares. It is also pretty easy if you can reduce the fraction to perfect squares.
If the numerator and denominator are not perfect squares, it is a little bit harder because you have to rationalize the denominator and simplify the square root in the numerator. But if you need a refresher on those topics, you can check out the linked pages to make it easier to find the square root of a fraction.
How to Find the Square Root of a Fraction
- Apply the square root to the numerator and the denominator separately.
- Simplify or rationalize the denominator.
- If the denominator is a perfect square, simplify it.
- Otherwise, rationalize the denominator.
- Simplify the numerator.
- Reduce the fraction, if possible.
Simplify:
\[\sqrt{\frac{4}{9}}\]
What is the square root of \(\frac{4}{9}\)?
\[\sqrt{\frac{4}{9}}=\frac{\sqrt{4}}{\sqrt{9}}\]
Is the denominator a perfect square?
Yes, the square root of 9 is 3. So, I can simplify the denominator.
\[\frac{\sqrt{4}}{\sqrt{9}}=\frac{\sqrt{4}}{3}\]
What is the most simplified form of the numerator?
4 is a perfect square, so I can simplify the square root completely.
\[\frac{\sqrt{4}}{\sqrt{9}}=\frac{2}{3}\]
Can the fraction be reduced?
No, 2 and 3 are both prime numbers so I can’t reduce \(\frac{2}{3}\).
\[\sqrt{\frac{4}{9}}=\frac{2}{3}\]
Simplify:
\[\sqrt{\frac{8}{5}}\]
What is the square root of \(\frac{8}{5}\)?
\[\sqrt{\frac{8}{5}}=\frac{\sqrt{8}}{\sqrt{5}}\]
Is the denominator a perfect square?
No, 5 is not a perfect square. So, I need to rationalize the denominator.
\[\frac{\sqrt{8}}{\sqrt{5}}\times {\red \frac {\sqrt{5}}{\sqrt{5}}}=\frac{\sqrt{40}}{\sqrt{25}}\]
The square root of 25 is 5, so the rationalized denominator is 5.
\[\frac{\sqrt{40}}{\sqrt{25}}=\frac{\sqrt{40}}{5}\]
What is the most simplified form of the numerator?
40 is not a perfect square. However, one of its factors (4) is a perfect square so it can be simplified.
\[\frac{\sqrt{40}}{5}=\frac{\sqrt{4}\sqrt{10}}{5}=\frac{2\sqrt{10}}{5}\]
Can the fraction be reduced?
No, 2 and 5 are both prime numbers so I can’t reduce \(\frac{2\sqrt{10}}{5}\).
Even though the 10 and the 5 share a common factor, I can’t reduce them because the 10 is inside a square root and the 5 is not.
\[\sqrt{\frac{8}{5}}=\frac{2\sqrt{10}}{5}\]
Simplify:
\[\sqrt{\frac{30}{32}}\]
What is the square root of \(\frac{30}{32}\)?
\[\sqrt{\frac{30}{32}}=\frac{\sqrt{30}}{\sqrt{32}}\]
Is the denominator a perfect square?
No, 32 is not a perfect square. So, I need to rationalize the denominator.
\[\frac{\sqrt{30}}{\sqrt{32}}\times {\red \frac {\sqrt{32}}{\sqrt{32}}}=\frac{\sqrt{960}}{\sqrt{1024}}\]
The square root of 1024 is 32, so the rationalized denominator is 32.
\[\frac{\sqrt{960}}{\sqrt{1024}}=\frac{\sqrt{960}}{32}\]
What is the most simplified form of the numerator?
960 is not a perfect square. However, one of its factors (64) is a perfect square so it can be simplified.
\[\frac{\sqrt{960}}{32}=\frac{\sqrt{64}\sqrt{15}}{32}=\frac{8\sqrt{15}}{32}\]
Can the fraction be reduced?
Yes, 8 and 32 are both divisible by 8 so I can reduce \(\frac{8\sqrt{15}}{32}\).
\[\frac{8\sqrt{15}}{32}=\frac{\sqrt{15}}{4}\]
\[\sqrt{\frac{30}{32}}=\frac{\sqrt{15}}{4}\]
Reduce in Advance
Reducing is the last step when you find the square root of a fraction because you should always make sure your answer is fully simplified.
However, in some cases, it can be easier to simplify the square roots if you reduce the fraction at the very beginning.
In the yellow example above, I found that \(\sqrt{\frac{30}{32}}=\frac{\sqrt{15}}{4}\).
In the process of finding that answer, I had to rationalize a denominator, recognize that 1024 was a perfect square, and simplify the square root of 960.
If I had reduced the fraction before I started Step 1, the entire process would have been much easier and I would have ended up with the same answer, as shown in this purple example.
Even if you reduce in advance, you should still double check your final answer to make sure it is fully reduced.
Simplify:
\[\sqrt{\frac{30}{32}}\]
Can the fraction be reduced?
Yes, 30 and 32 are both divisible by 2, so I can reduce \(\frac{30}{32}\).
\[\sqrt{\frac{30}{32}}=\sqrt{\frac{15}{16}}\]
What is the square root of \(\frac{15}{16}\)?
\[\sqrt{\frac{15}{16}}=\frac{\sqrt{15}}{\sqrt{16}}\]
Is the denominator a perfect square?
Yes, the square root of 16 is 4. So, I can simplify the denominator.
\[\frac{\sqrt{15}}{\sqrt{16}}=\frac{\sqrt{15}}{4}\]
What is the most simplified form of the numerator?
15 is not a perfect square and it does not have any factors that are perfect squares. So, I can’t simplify \(\sqrt{15}\) any further.
Can the fraction be reduced?
No, I can’t simplify \(\frac{15}{4}\) any further because 15 is inside a square root and 4 is not.
\[\sqrt{\frac{30}{32}}=\frac{\sqrt{15}}{4}\]
Why It Works
There are two reasons why we can apply the square root of a fraction to the numerator and denominator separately. The first reason is because of how squares and square roots are related to each other. The second reason is because of the exponent rules, specifically the distributive property of exponents and the fractional exponent rule.
Squares and Square Roots
Why can we apply the square root of a fraction to the numerator and denominator separately? \[\sqrt{\frac{16}{25}}=\frac{\sqrt{16}}{\sqrt{25}}\]
The square root of a fraction can be applied to the numerator and denominator separately because…
- Squares and square roots are inverse operations.
- The numerator and denominator of a fraction can be squared separately to find the square of a fraction.
- Square roots “un-do” squaring so the numerator and denominator can be square rooted separately to find the square root of a fraction.
Squares and square roots are inverse operations, which means that they “un-do” each other.
For example, if we square 5 it equals 25.
\[5^2=25\]
And if we square root 25 it equals 5.
\[\sqrt{25}=5\]
When we square a fraction, we multiply the fraction by itself.
\[\left( \frac{4}{5} \right)^{2} = \frac{4}{5} \times \frac{4}{5}\]
To multiply the fractions, we need to multiply the numerators and then multiply the denominators.
\[\frac{4}{5} \times \frac{4}{5}=\frac{16}{25}\]
Notice how the numerator and denominator are perfect squares. So, we could say…
\[\left( \frac{4}{5} \right)^{2} =\frac{4^2}{5^2}\]
This shows us that squaring a fraction is the same thing as squaring the numerator and denominator individually.
When we square fractions, we can square the numerator and denominator separately because of the way fraction multiplication works.
When we calculate the square root of a fraction, we can square root the numerator and denominator separately because we are just “un-doing” the fraction multiplication.
So, that is why…
\[\sqrt{\frac{16}{25}}=\frac{\sqrt{16}}{\sqrt{25}}\]
Exponent Rules
Why can we apply the square root of a fraction to the numerator and denominator separately? \[\sqrt{\frac{16}{25}}=\frac{\sqrt{16}}{\sqrt{25}}\]
The square root of a fraction can be applied to the numerator and denominator separately because…
- Roots can be written as fractional exponents.
- Those fractional exponents can be distributed to each part of the fraction
- Then the fractional exponents can be converted back to roots.
Any root can be re-written as a fractional exponent. A square root is the same thing as an exponent of \(\frac{1}{2}\).
\[\sqrt{\frac{16}{25}}=\left( \frac{16}{25} \right)^{1/2}\]
The distributive property of exponents says that exponents can be distributed across multiplication or division in parentheses.
\[\left( \frac{16}{25} \right)^{1/2}=\frac{16^{1/2}}{25^{1/2}}\]
The \(\frac{1}{2}\) exponents can be re-written as square roots.
\[\frac{16^{1/2}}{25^{1/2}}=\frac{\sqrt{16}}{\sqrt{25}}\]
So, that is why…
\[\sqrt{\frac{16}{25}}=\frac{\sqrt{16}}{\sqrt{25}}\]
