There are times when we need to simplify a square root such as √500 into a simpler form, that is 10√5. Similarly, the form can be simplified to . Additionally, the radical fraction can be simplified to (√5 – √3). This post discusses how to convert such forms into simpler ones.

**Simplifying √ a if some factor of a is a perfect square; a ≥ 0**

Express a as the product of factors some of which are perfect squares. Then, pull the roots of the perfect squares out of the radical sign. Suppose that a = a_{1}^{2}⋅b where b does not have any perfect square (other than 1) as its factor. Then √a = a_{1}√b.

**Example 1**

Simplify √500.

**Answer**

500 can be expressed as the product of factors some of which is a perfect square. It is the product of 100 (a perfect square) and 5. In short, 500 = 10^{2}⋅5. Pulling the root of 10^{2} out of the radical sign results in:

Thus, √500 = 10√5.

**Example 2**

Simplify √216.

**Answer**

216 can be expressed as the product of factors some of which is a perfect square. It is the product of 36 (a perfect square) and 6, i.e. 216 = 6^{2}⋅6. Pulling the root of 6^{2} out of the radical sign yields:

So, √216 = 6√6.

**Simplifying the form of ; ab > 0**

Multiply the numerator and denominator of by b so that the denominator of the fraction under the radical sign is a perfect square, i.e. b^{2}. We can pull the root of the perfect square out of the radical sign as follows.

**Example 3**

Simplify .

**Answer**

Multiply the numerator and denominator of by 3 so that the denominator of the fraction under the radical sign is a perfect square, i.e. 3^{2}. Pull the root of the perfect square out of the radical sign as follows.

.

Therefore, .

**Example 4**

Simplify .

**Answer**

Multiply the numerator and denominator of by 7 so that the denominator of the fraction under the radical sign is a perfect square, i.e. 7^{2}. Pull the root of the perfect square out of the radical sign as follows.

.

In conclusion, .

**Simplifying the form of ; b > 0**

Multiply the numerator and denominator of by √b. This results in:

**Example 5**

Simplify .

**Answer**

Multiply the numerator and denominator of by √11. It follows that:

**Example 6**

Simplify .

**Answer**

Multiply the numerator and denominator of by √5 to get:

**Simplifying the form of or **

To simplify a radical fraction of such forms, we multiply the numerator and denominator by the conjugate of the denominator. The conjugate of (√b + √c) is (√b – √c), whereas the conjugate of (√b – √c) is (√b + √c). As a consequence, to simplify the form , we multiply both the numerator and denominator by the conjugate of (√b + √c ), i.e. (√b – √c). The technique is demonstrated as follows.

**Example 7**

Simplify .

**Answer**

The conjugate of (√7 + √6) is (√7 – √6). To simplify the radical fraction, multiply it by . It follows that:

**Example 8**

Simplify .

**Answer**

The conjugate of (√5 + √2) is (√5 – √2). To simplify the fraction, multiply it by . It follows that:

To simplify the form , multiply both the numerator and denominator by the conjugate of (√b – √c ), that is (√b + √c). Hence:

**Example 9**

Simplify .

**Answer**

The conjugate of (√7 – √6) is (√7 + √6). To simplify the fraction, multiply it by . This results in:

**Example 10**

Simplify .

**Answer**

The conjugate of (2√3 – 3) is (2√3 + 3). To simplify the fraction, multiply it by . Therefore: