How to find the square root of a fraction

Related Pages
Rationalize The Denominator
Rationalising Surds

Separate Numerator & Denominator

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.

Reduce Fraction

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:
\(\sqrt {\frac{{18}}{{50}}} = \sqrt {\frac{9}{{25}}} = \frac{{\sqrt {{3^2}} }}{{\sqrt {{5^2}} }} = \frac{3}{5}\)

Change to Improper Fraction

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:

How to simplify fractions inside a square root?

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How to simplify square roots in the denominator of a fraction?

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Square Roots of Fractions/Rational Numbers

Finding the square of rational numbers for perfect squares as well as estimating non-perfect squares.

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Quotient Rule & Simplifying Square Roots

An introduction to the quotient rule for square roots and radicals and how to use it to simplify expressions containing radicals.

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RESULT

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DESCRIPTIONS

Step 1 Convert the Expression into Fraction Form

The square root of a fraction is equal to the square root of the numerator divided by the square root of the denominator.

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Step 2 Find the Simplified Forms of the Numerator and Denominator

• Numerator:

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• Denominator:

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We replace the numerator and denominator with their simplest forms.

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Step 3 Simplify the Radicands

3 and 2 have no common factors greater than 1. Thus, the radicands cannot be simplifed.

Step 4 Rationalize the Denominator

To rationalize the denominator, we multiply both the numerator and denominator by ....

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Step 5 Simplify the Fraction

The greatest common factor of 2 and 10 is 2. To simplify the expression, we can divide both the numerator and denominator by 2.

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Video transcript

- [Voiceover] So we have here the square root, the principal root, of one two-hundredth. And what I want to do is simplify this. When I say "simplify it" I really mean, if there's any perfect squares here that I can factor out to take it out from under the radical. And so I encourage you to pause the video and see if you can do that. Alright so there's a couple of ways that you could approach this. One way is to say, Well this is going to be the same thing as the square root of one over the square root of 200. The square root of one is just one over the square root of 200. And there's a couple of ways to try to simplify the square root of 200. I'll do it a couple of ways here. Square root of 200. You could realize that, OK, look 100 is a perfect square. And it goes into 200. So this is the same thing as two times 100. And so the square root of 200 is the square root of two times 100, which is the same thing as the square root of two times the square root of 100. And we know that the square root of 100 is 10. So it's the square root of two times 10 or we could write this as 10 square roots of two. That's one way to approach it. But if it didn't jump out at you immediately that you have this large perfect square that is a factor of 200, you could just start with small numbers. You could say, alright, let me do this alternate method in a different color. You could say, ah it's the same color that I've been doing before. (laughing) You could say that the square root of 200, say Well it's divisible by two. So it's two times 100. And if 100 didn't jump out at you as a perfect square, you could say, Well that's just going to be two times 50. Well I can still divide two into that. That's two times 25. Let's see, and 25, if that doesn't jump out at you as a perfect square, you could say that that's not divisible by two, not divisible by three, four, but it is divisible by five. That is five times five. And to identify the perfect squares you would say, Alright, are there any factors where I have at least two of them? Well I have two times two here. And I also have five times five here. So I can rewrite the square root of 200 as being equal to the square root of two times two. Let me just write it all out. Actually I think I'm going to run out of space. So the square root, give myself more space under the radical, square root of two times two times five times five times two. And I wrote it in this order so you can see the perfect squares here. Well this is going to be the same thing as the square root of two times two. This second method is a little bit more monotonous, but hopefully you see that it works, (laughing) I guess is one way to think about it. And they really, they boil down to the same method. We're still going to get to the same answer. So square root of two times two times the square root times the square root of five times five, times the square root of two. Well the square root of two times two is just going to be, this is just two. Square root of five times five, well that's just going to be five. So you have two times five times the square root of two, which is 10 times the square root of two. So this right over here, square root of 200, we can rewrite as 10 square roots of two. So this is going to be equal to one over 10 square roots of two. Now some people don't like having a radical in the denominator and if you wanted to get rid of that, you could multiply both the numerator and the denominator by the square root of two. 'Cause notice we're just multiplying by one, we're expressing one as square root of two over square root of two, and then what that does is we rewrite this as the square root of two over 10 times the square root of two times the square root of two. Well the square root of two times the square root of two is just going to be two. So it's going to be 10 times two which is 20. So it could also be written like that. So hopefully you found that helpful. In fact, even this one, you could write if you want to visualize it slightly differently, you could view it as one twentieth times the square root of two. So these are all the same thing.

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