Simplifying Square Roots (or Radicals)

Simplifying square roots involves expressing them in their simplest radical form. This means removing any perfect square factors from under the radical symbol (√).

In this lesson, we will learn two primary methods to simplify square roots:

1. Perfect Square Method
2. Prime Factorization Method

The following examples show how to simplify square roots: Find Perfect Square, Find Prime Factors. Scroll down the page for examples and solutions.

Simplify square roots using the perfect square method

The perfect square method for simplifying square roots is a direct and efficient way to do so when you can easily identify perfect square factors of the radicand (the number under the square root).

The steps involved are:

1. Find the largest perfect square that will divide the number in the square root.
2. Write the number as the product of the perfect square factor and the remaining factor.
3. Apply the Product Rule of Square Roots: \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\)
4. Take the square root of the perfect square factor (which will be a whole number) and write it outside the radical.
5. The factor that is not a perfect square remains under the square root.

Example:
Simplify \(\sqrt{48}\)

Solution:
Step 1: The perfect square 16 divides 48.
Step 2: Write 48 as the product of 16 and 3.
48 = 16 × 3
Step 3: Apply the product rule and simplify the perfect square. \(\sqrt{48} = \sqrt{16 \times 3} = \sqrt{4 \times 4 \times 3} = 4\sqrt{3}\)

How to simplify square roots using the perfect square method?
The following video shows more examples of simplifying square roots using the perfect square method. The perfect square method is suitable for small numbers for example less than 1000. For bigger numbers the prime factorization method may be better.

It would be useful for you to memorize the first five perfect squares of prime numbers.
1² = 1, 2² = 4, 3²= 9, 5² = 25, 7² = 49, 11² = 121

Step 1: Factor out the perfect squares
Step 2: Separate perfect squares using product of square roots property
Step 3: Simplify

Examples:
Simplify the following square roots:
a) square root of 72
b) square root of 288
c) square root of 108

• Show Video Lesson

How to simplify square roots by factoring out perfect squares?
Example:
Simplify the following square roots:
a) square root of 60
b) square root of 108

• Show Video Lesson

Simplify square roots using the prime factorization method

The steps involved are:

1. Break the number in the square root into prime factors.
   
2. For each pair of factors, “take one out” of the square root sign.
   
3. The remaining factors in the square root sign are multiplied together.

Example:
Simplify \(\sqrt{48}\)

Solution:
Step 1. Break the number 48 into prime factors
48 = 2 × 2 × 2 × 2 × 3
Step 2: For each pair of 2’s, take one outside the square root symbol.
\(\sqrt{48} = 2 \times 2 \times \sqrt{3} = 4\sqrt{3}\)

Example:
Simplify \(\sqrt{90}\)

Solution:
Step 1. Break the number 90 into prime factors
90 = 2 × 3 × 3 × 5
Step 2: Take 3 out of the square root sign
Step 3: Multiply 2 and 5
\(\sqrt{90} = 3\sqrt{2 \times 5} = 3\sqrt{10}\)

How to simplifying square roots using the prime factorization method?
The following video shows more examples of simplifying square roots using the prime factorization method.
Step 1: Factor into product of primes
Step 2: Circle the pairs of factors
Step 3: Remove the pairs and multiply by each number removed.

Example:
Simplify the following square roots:
a) square root of 18
b) square root of 420

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Examples of simplifying square roots using the prime factorization.
Example:
Simplify the following square roots:
a) square root of 180
b) square root of 200

• Show Video Lesson

How to use prime factorization to simplify square roots?
Example:
Simplify the following square roots:
a) square root of 84
b) square root of 392

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How to Simplify Square Roots With Fractions?
How to deal with square roots in the denominator of a fraction.

Example:
Simplify the following square roots:
a) \(\sqrt {\frac{7}{3}} \)
b) \(\sqrt {\frac{24}{5}} \)

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Which Method to Use?

• Perfect Square Factor: This method can be faster if you can easily identify the perfect square factor. However, if you don’t find the largest one, you might need to simplify further.
• Prime Factorization: This method is more systematic and always works, especially for larger numbers where it might be harder to identify the largest perfect square factor.

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