Rationalize the denominator. 2 5 \frac{\sqrt{2}}{\sqrt{5}} 5 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"> 2 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z">

Hello, today we are going to be rationalizing the denominator of the radical expression. The radical expression given to us is the square root of seven divided by the square root of 37. So how can we rationalize a denominator? While in order to rationalize a denominator, we want to simplify the denominator in a way that allows us to remove the square root symbol. Recall that the square root symbol allows us to simplify perfect squares into individual terms. So if we have the square root of 37 in the denominator, we can multiply the square root by itself in order to eliminate the square root symbol. But if we multiply the denominator by the square root of 37 we must also multiply the numerator by the square root of 37. So let's go ahead and take the products of the numerator and denominator. In the numerator, we will have the square root of seven multiplied by 37 which will give us the square root of 259. And in the denominator, we will have the square root of 37 multiplied by the square root of 37 which will simplify to give us the square root of 37 squared. Notice that in the denominator, we have a perfect square inside the square root symbol using the property that I stated in the upper right. We can simplify this denominator to just be 37. So we are left with the square root of 259 divided by 37. This is going to be the simplest form of the given rational expression with a rationalized denominator. And with that being said, the solution to this problem is going to be c so I hope this video helps you in understanding how to rationalize a denominator. And I'll go ahead and see you all in the next video.

0. Review of Algebra

Rationalize Denominator

College Algebra

0. Review of Algebra

Rationalize Denominator

2:09 minutes

Problem 47

Textbook Question

Rationalize the denominator.
$\frac{\sqrt{2}}{\sqrt{5}}$

Verified step by step guidance

Identify the expression that needs rationalization: $ \frac{\sqrt{2}}{\sqrt{5}} $. The goal is to eliminate the square root in the denominator.

Multiply both the numerator and the denominator by $ \sqrt{5} $ to rationalize the denominator. This uses the property that $ \sqrt{a} \times \sqrt{a} = a $.

The expression becomes $ \frac{\sqrt{2} \times \sqrt{5}}{\sqrt{5} \times \sqrt{5}} $.

Simplify the denominator: $ \sqrt{5} \times \sqrt{5} = 5 $.

Simplify the numerator: $ \sqrt{2} \times \sqrt{5} = \sqrt{10} $. The rationalized expression is $ \frac{\sqrt{10}}{5} $.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Rationalizing the Denominator

Rationalizing the denominator involves eliminating any irrational numbers from the denominator of a fraction. This is typically done by multiplying both the numerator and the denominator by a suitable expression that will result in a rational number in the denominator. For example, to rationalize \\frac{\sqrt{2}}{\sqrt{5}}\, one would multiply by \\frac{\sqrt{5}}{\sqrt{5}}\, resulting in \\frac{\sqrt{10}}{5}\.

Properties of Square Roots

Understanding the properties of square roots is essential for manipulating expressions involving them. Key properties include \\sqrt{a} \\cdot \\sqrt{b} = \\sqrt{a \\cdot b} and \\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}. These properties allow for simplification and rationalization of expressions, making it easier to work with square roots in algebraic contexts.

Simplifying Fractions

Simplifying fractions is the process of reducing a fraction to its simplest form, where the numerator and denominator have no common factors other than 1. This is important in rationalizing denominators, as it ensures that the final expression is as concise and clear as possible. For instance, after rationalizing \\frac{\sqrt{2}}{\sqrt{5}}\, one should check if the resulting fraction can be simplified further.

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