Calculating the square root of a mixed fraction might seem complex at first, but it can be simplified with a clear method. A mixed fraction is a combination of a whole number and a proper fraction. To find its square root, the mixed fraction is first converted to an improper fraction, which makes the calculation more straightforward.

In this article, we will discuss mixed fractions and how to calculate their square roots, whether the numerator and denominator are perfect squares or not.

What are Mixed Fractions?

Mixed fractions, also known as mixed numbers, are numbers that consist of a whole number and a proper fraction combined.

For example, [Tex]1\frac{3}{4}[/Tex]​ is a mixed fraction where 1 is the whole number and 3/4​ is the proper fraction.

Mixed fractions are often used to represent quantities that are more than one but less than two, three, etc., offering a convenient way to express parts of a whole.

Read More about Mixed Fractions.

Square Root of Mixed Fraction

Finding the square root of a mixed fraction involves a few straightforward steps. A mixed fraction combines a whole number and a proper fraction, such as [Tex]2\frac{4}{7}[/Tex].

How to Find Square Root of Mixed Fraction?

To find square root of any mixed fraction, we can use the following steps:

Step 1: Convert the mixed fraction to an improper fraction, which includes the following substeps.

• Multiply the whole number part by the denominator of the fraction part.
• Add the numerator of the fraction part to this product.
• Place this sum over the original denominator.

For example, [Tex]1\frac{3}{4}[/Tex] converts to an improper fraction as follows:

[Tex]1\frac{3}{4} = \frac{1 \times 4 + 3}{4} = \frac{7}{4}[/Tex]

Step 2: Take the square root of the numerator and the square root of the denominator separately.

Using the improper fraction 7/4:

[Tex]\sqrt{\frac{7}{4}} = \frac{\sqrt{7}}{\sqrt{4}} = \frac{\sqrt{7}}{2}[/Tex]

Step 3: If the square roots can be simplified further, do so. If not, leave it in its simplest form.

Since √7 is already in its simplest form, [Tex]\frac{\sqrt{7}}{2}[/Tex] is the simplified form of the square root of [Tex]1\frac{3}{4}[/Tex].

Note: We can further calculate square root in decimals using approximation i.e., √7 ≈ 2.65.

For Perfect Square Numerator and Denominator

Consider the fraction i.e., [Tex]1\frac{16}{9}[/Tex] and find it’s square root.

Step 1: Convert [Tex]1\frac{16}{9}[/Tex]:

[Tex]1\frac{16}{9} = \frac{1 \times 9 + 16}{9} = \frac{9 + 16}{9} = \frac{25}{9}[/Tex]

Step 2: Now, find the square root of 25/9:

Here both numerator and denominator are both perfect square.

[Tex]\sqrt{\frac{25}{9}} = \frac{\sqrt{25}}{\sqrt{9}} = \frac{5}{3}[/Tex]

For Non-Perfect Square Numerator and Denominator

Consider the fraction i.e., [Tex]2\frac{7}{5}[/Tex] and find it’s square root.

Step 1: Convert the Mixed Fraction to an Improper Fraction:

[Tex]2\frac{7}{5} = \frac{2 \times 5 + 7}{5} = \frac{10 + 7}{5} = \frac{17}{5}[/Tex]

Step 2: Since 17 and 5 are not perfect squares, we find the square root of 17/5 by using the square root property for fractions.

[Tex]\sqrt{\frac{17}{5}} = \frac{\sqrt{17}}{\sqrt{5}}[/Tex]

Step 3: [Tex]\frac{\sqrt{17}}{\sqrt{5}} [/Tex]can be simplified by rationalizing the denominator. To rationalize the denominator, multiply both the numerator and the denominator by √5:

[Tex]\frac{\sqrt{17}}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{\sqrt{17} \cdot \sqrt{5}}{5} = \frac{\sqrt{85}}{5}[/Tex]

Read More,

• Fractions
• Comparing Fractions
• How to Divide Mixed fractions?
• Subtracting Mixed Fractions

Solved Examples on Square Root of Mixed Fraction

Example 1: Find the square root of 2 1/4.

Solution:

2 1/4 = (2 × 4 + 1) / 4 = 9 / 4

Thus, √(9 / 4) = √9 / √4 = 3 / 2

∴The square root of 2 1/4 is 3/2.

Example 2: Find the square root of 3 1/9.

Solution:

3 1/9 = (3 × 9 + 1) / 9 = 28 / 9

Thus, √(28 / 9) = √28 / √9 = √(4 × 7) / 3 = (2 × √7) / 3

∴The square root of 3 1/9 is (2 × √7) / 3.

Example 3: Find the square root of 1 4/9.

Solution:

1 4/9 = (1 × 9 + 4) / 9 = 13 / 9

Thus, √(13 / 9) = √13 / √9 = √13 / 3

∴The square root of 1 4/9 is √13 / 3.

Example 4: Find the square root of 2 1/25.

Solution:

2 1/25 = (2 × 25 + 1) / 25 = 51 / 25

Thus, √(51 / 25) = √51 / √25 = √51 / 5

∴The square root of 2 1/25 is √(51) / 5.

Example 5: Find the square root of 4 1/2.

Solution:

4 1/2 = (4 × 2 + 1) / 2 = 9 / 2

Thus, √(9 / 2) = √9 / √2 = 3 / √2 = (3 × √2) / 2

∴The square root of 4 1/2 is (3 × √2) / 2.

Practice Problems on Square Root of Mixed Fraction

Problem 1: Find the square root of [Tex]6 \frac{1}{4}[/Tex]​.

Problem 2: Calculate the square root of [Tex]3 \frac{1}{16}[/Tex]​.

Problem 3: Determine the square root of [Tex]2 \frac{1}{4}[/Tex]​.

Problem 4: What is the square root of [Tex]2 \frac{55}{49}[/Tex]?

Problem 5: Find the square root of [Tex]2 \frac{14}{25}[/Tex].

Problem 6: Calculate the square root of [Tex]1 \frac{48}{121}[/Tex]​.

Problem 7: Determine the square root of [Tex]1 \frac{52}{144}[/Tex]​.

Problem 8: What is the square root of [Tex]2 \frac{56}{81}[/Tex]​?

Problem 9: Find the square root of [Tex]1 \frac{87}{169}[/Tex]​.

Problem 10: Calculate the square root of [Tex]1 \frac{64}{225}[/Tex]​.

FAQs on Square Root of Mixed Fraction

What is a mixed fraction?

A mixed fraction is a combination of a whole number and a proper fraction. For example, [Tex]2 \frac{1}{2}[/Tex]​ is a mixed fraction where 2 is the whole number and 1/2​ is the proper fraction.

How do you calculate the square root of a mixed fraction?

To calculate the square root of a mixed fraction, first convert the mixed fraction to an improper fraction, then find the square root of the numerator and the denominator separately. If both are perfect squares, simplify the result back to a mixed fraction if needed.

What if the numerator and denominator are not perfect squares?

If the numerator and denominator are not perfect squares, you can only approximate the square root. Simplify the square root expression as much as possible, but the result will often be an irrational number.

Is it necessary for the numerator and denominator to be perfect squares?

No, it is not necessary. However, if both the numerator and denominator are perfect squares, the square root will be a simpler, rational number.

How do you convert a mixed fraction to an improper fraction?

To convert a mixed fraction to an improper fraction, multiply the whole number by the denominator of the fraction part, then add the numerator of the fraction part. Write the result over the original denominator.