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Radicals that have Fractions – 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√a = a 1/n,
where n is referred to as the index and a is the radicand, then the symbol √ is called the 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
Let’s explain this technique with the help of the example below.
Example 1
Simplify the following expression:
√27/2 x √(1/108)
Solution
Two radical fractions can be combined by following these relationships:
√a / √b = √(a / b) and √a x √b =√ab
Therefore,
√27/2 x √(1/108)
= √27/√4 x √(1/108)
= √(27 / 4) x √(1/108)
= √(27 / 4) x √(1/108) = √(27 / 4 x 1/108)
= √(27 / 4 x 108)
Since 108 = 9 x 12 and 27 = 3 x 9
√(3 x 9/ 4 x 9 x 12)
9 is a factor of 9, and so simplify,
√(3 / 4 x 12)
= √(3 / 4 x 3 x 4)
= √(1 / 4 x 4)
=√(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 √2 and √3 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 / √2
Solution
Multiply both the numerator and denominator by the root of 2.
= (1 / √2 x √2 / √2)
= √2 / 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.
Example 3
Rationalize the expression:1 / (3 − √2)
Solution
Multiply both the top and bottom by the (3 + √2) as the conjugate.
1 / (3 − √2) x (3 + √2) / (3 + √2)
= (3 + √2) / (3 2 – (√2) 2)
= (3 + √2) / 7, the denominator is now rational.
Example 4
Rationalize the denominator of the expression; (2 + √3)/(2 – √3)
Solution
- In this case, 2 – √3 is the denominator and rationalizes the denominator, both top and bottom by its conjugate.
The conjugate of 2 – √3 = 2 + √3.
- Comparing the numerator (2 + √3) ² with the identity (a + b) ²= a ²+ 2ab + b ², the result is 2 ² + 2(2)√3 + √3² = (7 + 4√3)
- Comparing the denominator with the identity (a + b) (a – b) = a ² – b ², the results is 2² – √3²
Example 5
Rationalize the denominator of the following expression,
(5 + 4√3)/(4 + 5√3)
Solution
- 4 + 5√3 is our denominator, and so to rationalize the denominator, multiply the fraction by its conjugate; 4+5√3 is 4 – 5√3
- Multiplying the terms of the numerator; (5 + 4√3) (4 – 5√3) gives out 40 + 9√3
- Compare the numerator (2 + √3) ² the identity (a + b) ²= a ²+ 2ab + b ², to get
4 ²- (5√3) ² = -59
Example 6
Rationalize the denominator of (1 + 2√3)/(2 – √3)
Solution
- We have 2 – √3 in the denominator, and to rationalize the denominator, multiply the entire fraction by its conjugate
Conjugate of 2 – √3 is 2 + √3
- We have (1 + 2√3) (2 + √3) in the numerator. Multiply these terms to get, 2 + 6 + 5√3
- Compare the denominator (2 + √3) (2 – √3) with the identity
a ²- b ² = (a + b) (a – b), to get 2 ² – √3 ² = 1
Example 7
Rationalize the denominator,
(3 + √5)/(3 – √5) + (3 – √5)/(3 + √5)
Solution
- Find the LCM to get (3 +√5)² + (3-√5)²/(3+√5)(3-√5)
- Expand (3 + √5) ² as 3 ² + 2(3)(√5) + √5 ² and (3 – √5) ² as 3 ²- 2(3)(√5) + √5 ²
Compare the denominator (3-√5)(3+√5) with identity a ² – b ²= (a + b)(a – b), to get
3 ² – √5 ² = 4
Example 8
Rationalize the denominator of the following expression:
[(√5 – √7)/(√5 + √7)] – [(√5 + √7) / (√5 – √7)]
Solution
- By calculating the L.C.M, we get
(√5 – √7) ² – (√5 + √7) ² / (√5 + √7)(√5 – √7)
- Expansion of (√5 – √7) ²
= √5 ² + 2(√5)(√7) + √7²
- Expansion of (√5 + √7) ²
= √5 ² – 2(√5)(√7) + √7 ²
- Compare the denominator (√5 + √7)(√5 – √7) with the identity
a² – b ² = (a + b)(a – b), to get
√5 ² – √7 ² = -2