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# Radicals that have Fractions â€“ Simplification Techniques

A radical can be defined as a symbol that indicates the root of a number. Square root, cube root, fourth root are all radicals. This article introduces by defining common terms in fractional radicals. If *n* is a positive integer greater than 1 and *a* is a real number, then;

^{n}âˆša = a^{ 1/n},

where *n* is referred to as the index and *a* is the radicand, then the symbolÂ âˆšÂ is called theÂ **radical**. The right and left side of this expression is called exponent and radical form respectively.

## How to Simplify Fractions with Radicals?

- Simplifying a radical by factoring out.
- Rationalizing the fraction or eliminating the radical from the denominator.

### Simplifying Radicals by Factoring

Letâ€™s explain this technique with the help of the example below.

*Example 1*

Simplify the following expression:

âˆš27/2 x âˆš(1/108)

__Solution__

Two radical fractions can be combined by following these relationships:

âˆša / âˆšb = âˆš(a / b)Â and âˆša x âˆšb =âˆšab

Therefore,

âˆš27/2 x âˆš(1/108)

= âˆš27/âˆš4 x âˆš(1/108)

= âˆš(27 / 4) x âˆš(1/108)

= âˆš(27 / 4) x âˆš(1/108) = âˆš(27 / 4 x 1/108)

= âˆš(27 / 4 x 108)

Since 108 = 9 x 12 and 27 = 3 x 9

âˆš(3 x 9/ 4 x 9 x 12)

9 is a factor of 9, and so simplify,

âˆš(3 / 4 x 12)

= âˆš(3 / 4 x 3 x 4)

= âˆš(1 / 4 x 4)

=âˆš(1 / 4 x 4) = 1 / 4

### Simplifying Radicals by Rationalizing the Denominator

Rationalizing a denominator can be termed an operation where the root of an expression is moved from the bottom of a fraction to the top. The bottom and top of a fraction are called the denominator and numerator, respectively. Numbers such as 2 and 3 are rational, and roots such as âˆš2 and âˆš3 are irrational. In other words, a denominator should always be rational, and this process of changing a denominator from irrational to rational is what is termed as “Rationalizing the Denominator.”

There are two ways of rationalizing a denominator. A radical fraction can be rationalized by multiplying both the top and bottom by a root:

*Example 2*

Rationalize the following radical fraction: 1 / âˆš2

Solution

Multiply both the numerator and denominator by the root of 2.

= (1 / âˆš2 x âˆš2 / âˆš2)

= âˆš2 / 2

Another method of rationalizing the denominator is the multiplication of both the top and bottom by a conjugate of the denominator. A conjugate is an expression with a changed sign between the terms. For example, a conjugate of an expression such as x ^{2} + 2 is

x ^{2} – 2.

*Â *

*Example 3 *

Rationalize the expression:1 / (3 âˆ’ âˆš2)

__Solution__

Multiply both the top and bottom by the (3 + âˆš2) as the conjugate.

1 / (3 âˆ’ âˆš2) x (3 + âˆš2) / (3 + âˆš2)

= (3 + âˆš2) / (3^{ 2} – (âˆš2)^{ 2})

= (3 + âˆš2) / 7, the denominator is now rational.

*Example 4*

Rationalize the denominator of the expression; (2 + âˆš3)/(2 – âˆš3)

__Solution__

- In this case, 2 – âˆš3 is the denominator and rationalizes the denominator, both top and bottom by its conjugate.

The conjugate ofÂ 2 – âˆš3 =Â 2 + âˆš3.

- Comparing the numerator (2 + âˆš3) Â²Â with the identityÂ (a + b) Â²= a Â²+ 2ab + b Â², the result is 2 Â² + 2(2)âˆš3 +Â âˆš3Â² =Â Â (7 + 4âˆš3)
- Comparing the denominator with the identityÂ (a + b) (a – b) =Â a Â² – b Â², the results is 2Â² –Â âˆš3Â²

*Example 5*

Rationalize the denominatorÂ of the following expression,

(5 + 4âˆš3)/(4Â + 5âˆš3)

__Solution__

- 4 + 5âˆš3 is our denominator, and so to rationalize the denominator, multiply the fraction by its conjugate; 4+5âˆš3 isÂ 4 – 5âˆš3
- Multiplying the terms of the numerator; (5Â + 4âˆš3) (4 – 5âˆš3) gives out 40 + 9âˆš3
- Compare the numerator (2 + âˆš3) Â²Â the identityÂ (a + b) Â²= a Â²+ 2ab + b Â², to get

4 Â²- (5âˆš3) Â² =Â -59

*Example 6*

Rationalize the denominatorÂ of (1 + 2âˆš3)/(2 – âˆš3)

__Solution__

- We have 2 – âˆš3 in the denominator, and to rationalize the denominator, multiply the entire fraction by its conjugate

Conjugate ofÂ 2 – âˆš3Â isÂ 2 + âˆš3

- We have (1 + 2âˆš3) (2 + âˆš3) in the numerator. Multiply these terms to get, 2 + 6 + 5âˆš3
- Compare the denominator (2 + âˆš3) (2 – âˆš3)Â with the identity

a Â²- b Â² = (a + b) (a – b), to get 2 Â² – âˆš3 Â² = 1

*Â *

*Example 7*

Rationalize the denominator,

(3 + âˆš5)/(3Â – âˆš5) +Â (3 – âˆš5)/(3Â + âˆš5)

__Solution__

- Find the LCM to get (3 +âˆš5)Â² + (3-âˆš5)Â²/(3+âˆš5)(3-âˆš5)
- ExpandÂ (3 + âˆš5) Â² as 3 Â² + 2(3)(âˆš5) + âˆš5 Â² and Â (3 – âˆš5) Â² as 3 Â²- 2(3)(âˆš5) + âˆš5 Â²

Compare the denominator (3-âˆš5)(3+âˆš5)Â with identity a Â² – b Â²= (a + b)(a – b), to get

3 Â² – âˆš5 Â² = 4

*Example 8 *

Rationalize the denominator of the following expression:

[(âˆš5 – âˆš7)/(âˆš5 + âˆš7)] – [(âˆš5 + âˆš7) / (âˆš5Â – âˆš7)]

__Solution__

- By calculating the L.C.M, we get

(âˆš5 – âˆš7) Â² – (âˆš5 + âˆš7) Â² / (âˆš5 + âˆš7)(âˆš5 – âˆš7)

- Expansion ofÂ (âˆš5Â – âˆš7) Â²

= âˆš5 Â² + 2(âˆš5)(âˆš7) +Â âˆš7Â²

- Expansion ofÂ (âˆš5 + âˆš7) Â²

= âˆš5 Â² – 2(âˆš5)(âˆš7) +Â âˆš7 Â²

- Compare the denominatorÂ (âˆš5 +Â âˆš7)(âˆš5Â – âˆš7)Â with the identity

aÂ² – b Â²Â =Â (a + b)(a – b), to get

âˆš5 Â²Â – âˆš7 Â² = -2