Fractions with square roots can seem daunting at first, but with a little practice and understanding, they can become much easier to work with. In this blog post, we will explore what fractions with square roots are, how to simplify them, and provide examples to help you understand the process.
What are Fractions with Square Roots?
A fraction with a square root is a fraction where the numerator or denominator, or both, contain a square root. The square root symbol (√) indicates that the number under the root is the product of the same number multiplied by itself. For example, the square root of 9 is √9, which equals 3.
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Here are some examples of fractions with square roots:
√2/4
5/√3
√7/√2
Simplifying Fractions with Square Roots
Simplifying fractions with square roots is similar to simplifying regular fractions. The goal is to reduce the fraction to its simplest form. The steps to simplify a fraction with a square root are as follows:
Step 1: Simplify the Square Root
The first step is to simplify any square roots in the fraction. This means finding the largest perfect square that can be factored out of the square root. For example, the square root of 72 can be simplified as follows:
√72 = √(36 × 2) = √36 × √2 = 6√2
Step 2: Rationalize the Denominator
The second step is to rationalize the denominator. This means getting rid of any square roots in the denominator. To do this, we multiply the numerator and denominator by the conjugate of the denominator. The conjugate of a binomial is the same binomial, but with the opposite sign between the terms. For example, the conjugate of √3 + 2 is √3 – 2.
Here’s an example of rationalizing the denominator:
5/√3 = (5/√3) x (√3/√3) = 5√3/3
Note that multiplying by the conjugate eliminates the square root in the denominator because it creates a difference of squares.
Examples of Simplifying Fractions with Square Roots
Let’s take a look at some examples of simplifying fractions with square roots.
Example 1:
Simplify √18/3
Step 1: Simplify the Square Root
√18 = √(9 × 2) = 3√2
Step 2: Simplify the Fraction
√18/3 = (3√2)/3 = √2
As a result, √18/3 can be simplified to √2.
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Example 2:
Simplify 2/√12
Step 1: Simplify the Square Root
√12 = √(4 × 3) = 2√3
Step 2: Rationalize the Denominator
2/√12 = (2/√12) x (√12/√12) = 2√12/12 = √3/3
Therefore, the simplified form of 2/√12 is √3/3.
Example 3:
Simplify √6/√2
Step 1: Simplify the Square Roots
√6 = √(2 × 3) = √2 × √3
Step 2: Simplify the Fraction
√6/√2 = (√2 × √3)/√2 = √3
Therefore, the simplified form of √6/√2 is √3.
Practice Makes Perfect
The more practice you have with fractions containing square roots, the easier they will become. Here are some additional examples for you to try on your own:
Example 4:
Simplify (3√5)/5
Step 1: Simplify the Fraction
(3√5)/5 = 3/5 x √5
Therefore, the simplified form of (3√5)/5 is 3/5√5.
Example 5:
Simplify 4/√18
Step 1: Simplify the Square Root
√18 = √(9 × 2) = 3√2
Step 2: Rationalize the Denominator
4/√18 = (4/√18) x (√18/√18) = 4√18/18 = 2√2/3
Therefore, the simplified form of 4/√18 is 2√2/3.
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Summary
Fractions with square roots may seem challenging at first, but with practice and understanding, they can become much easier to simplify. Remember to simplify any square roots in the fraction and then rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator. The more practice you have with these types of fractions, the easier they will become. With these tools, you’ll be able to tackle any fraction with square roots that comes your way.