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Simplify by rationalizing the denominator of (7 + √6)/(3 – √2)

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  • Last Updated : 22 Aug, 2022
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The term “number system” refers to the representation of numbers, where a “number” is a mathematical value used in various mathematical operations such as counting, measuring, labeling, and computation. There are different types of numbers, such as natural numbers, whole numbers, integers, rational and irrational numbers, real numbers, etc. These numbers are used as digits in a number system. Similarly, a number system is classified into various types that have different properties, like a binary number system, an octal number system, a decimal number system, and a hexadecimal number system. 

Radicals are an expression with a root, such as a square root, a cube root, a fourth root, etc. If the index of a radical expression is not mentioned, the root is assumed to be a square root. The “n√”  is the radical symbol that means “nth root of”. For instance, the “nth root of (a-b)” is symbolically written as shown in the figure given below. Here, “n” refers to the index or degree, “(a-b)” is the radicand, and “(n√)’ is the radical symbol. The root of a whole number with an irrational value is called a surd. For example, √2, 5 + √3, 2√3, etc are some examples of surds.

 

Rationalization

Rationalization is the process of making the denominator of an algebraic fraction free from an irrational number. While performing the rationalization, we eliminate a surd or a radical from the denominator such that it only has a rational number. Having rational numbers in the denominators makes it very easy to compare values or perform mathematical calculations. So, to make the denominator a rational number, a rational factor or conjugate is multiplied by both the numerator and the denominator of the fraction.

Rationalizing a Denominator with One term

The process of multiplying a surd with another similar surd results in a rational number and this process is termed “rationalization.” The surd that is used for multiplying is known as the “rationalizing factor.” Let us consider a fraction that has a radical in the denominator, i.e., 1/√x. Now, to rationalize √x we need to multiply both the numerator and the denominator with √x.

Example: Rationalize: 8/3√2.

Solution:

Given: 8/3√2 = 8/(2)1/3

Now, multiply both numerator and denominator with (2)2/3.

= 8/(2)1/3 × (2)2/3/(2)2/3

= 8(2)2/3/2(1/3+2/3)

= 8(2)2/3/ 2 = 4(22)1/3

= 43√4

Hence, 8/3√2  = 43√4

Rationalizing a Denominator with two terms.

To rationalize a denominator, we need to learn about a conjugate, which is a similar surd but has a different sign. For example, the conjugate of (a+√b) is (a−√b). In mathematics, a number’s conjugate is a number that, when multiplied or added to the given number, yields a rational number. While performing the rationalization, both the numerator and denominator are multiplied with a suitable conjugate.

Example: Rationalize: 1/(4 − √3).

Solution:

Given: 1/(4 − √3)

To rationalize the denominator, multiply and divide the given term with (4 + √3).

= 1/(4 − √3) × (4 + √3)/(4 + √3)

Since, (a + b)(a – b) = a2 – b2

(4 − √3)×(4 + √3) = (4)2 − (√3)2 = 16 − 3 = 13

⇒ 1/(4 − √3) × (4 + √3)/(4 + √3) = (4 + √3)/13

 Thus, 1/(4 − √3) = (4 + √3)/13.

How to Rationalize the Denominator?

Follow the steps mentioned below to rationalize the denominator of a fraction.

Step 1: To remove the radicals in the denominator, multiply both the denominator and numerator of the given fraction with a suitable radical.

Step 2: Make sure that all surds in the fraction are in their simplified form.

Step 3: If necessary, simplify the fraction further.

Rationalize the denominator of (7 + √6)/(3–√2).

Solution:

Given expression: (7 + √6)/(3–√2).

To rationalize the denominator, multiply and divide the given term with (3 + √2).

= (7+√6)/(3–√2) × (3 + √2)/(3 + √2)

From algebraic formula, we have (a + b)(a – b) = a2 – b2

So, (3 – √2)×(3 + √2) = (3)2 – (√2)2

= 9 – 2 = 7

(7+√6)/(3–√2) × (3 + √2)/(3 + √2) = [(7+√6)(3+√2)]/7

= [7(3+√2) + √6(3+√2)]/7

= [21+7√2+3√6+√12]/7

= [21+7√2+3√6+2√3]/7

Hence, (7 + √6)/(3–√2) = (21+7√2+2√3+3√6)/7.

Solved Examples of Rationalization

Example 1: Rationalise: 21/√5.

Solution:

Given: 21/√5

To rationalize the denominator, multiply and divide the given term with √5.

= 21/√5× √5/√5

= 21√5/5

Hence,

21/√5 = 21√5/5.

Example 2: Rationalize: 6/(2 + √3).

Solution:

Given: 6/(2 + √3).

To rationalize the denominator, multiply and divide the given term with (2 − √3).

= 6/(2 + √3) × (2 − √3)/(2 − √3)

Since, (a + b)(a – b) = a2 – b2

(2 + √3)×(2 − √3) = (2)2 − (√3)2

= 4 − 3 = 1

6/(2 + √3) × (2 − √3)/(2 − √3)

= 6(2 − √3)/1

= 6(2 − √3)

Hence, 6/(2 + √3) = 6(2 − √3).

Example 3: Rationalize: √2/(√6 + √11).

Solution:

Given: √2/(√6 + √11)

To rationalize the denominator, multiply and divide the given term with (√6 − √11).

= √2/(√6 + √11) × (√6 − √11)/(√6 − √11)

Since, (a + b)(a – b) = a2 – b2

(√6 + √11)×(√6 − √11) = (√6)2 − (√11)2

= 6 − 11 = −5

√2/(√6 + √11) × (√6 − √11)/(√6 − √11) = √2(√6 − √11)/(−5)

= (√12 − √22)/(−5)

= (√22 − √12)/5

Hence, √2/(√6 + √11) = (√22 − √12)/5.

Example 4: Rationalize: 13/ 4√27.

Solution:

Given: 13/ 4√27

13/ 4√27. = 13/(27)(1/4)

To rationalize the denominator, multiply and divide the given term with (27)3/4.

=13/(27)(1/4) × (27)3/4/(27)3/4

= 13 × (27)3/4/27(1/4 + 3/4)

= 13 × (27)3/4/27

= 13/27 × (27)3/4

=  13/27 [4√(27)3]

Therefore, 13/ 4√27 = 13/27 [4√(27)3].

Example 5: Rationalize: (4 − √2)/(5 + √6).

Solution:

Given: (4 − √2)/(5 + √6)

To rationalize the denominator, multiply and divide the given term with (5 − √6).

= (4 − √2)/(5 + √6) × (5 − √6)/(5 − √6)

Since, (a + b)(a – b) = a2 – b2

(5 + √6)×(5 − √6) = (5)2 − (√6)2

= 25 − 6 = 19

(4 − √2)/(5 + √6) × (5 − √6)/(5 − √6) = (4 − √2)×(5 − √6)/19

= [4(5 − √6) − √2(5 − √6)]/19

= [20 − 4√6 − 5√2 − √12]/19

Hence, (4 − √2)/(5 + √6) = [20 − 4√6 − 5√2 − √12]/19.

FAQs on Rationalization of Denominator

Question 1: What is meant by “rationalization”?

Answer:

Rationalization is called the process of making a denominator of an algebraic fraction free from an irrational number. While performing the rationalization, we eliminate a surd or a radical from the denominator such that it only has a rational number.

Question 2: What is an example of rationalization?

Answer:

Any fraction that has an irrational number in the denominator can be rationalized to eliminate the radical from the denominator. 

Example: Rationalize: √(49/2)

Solution: 

Given: √(49/2) = √49/√2

= 7/√2.

Now, multiply the numerator and denominator by √2.

= 7/√2 × √2/√2

√(49/2) = 7√2/2.

Question 3: What are the rules followed while performing Rationalization?

Answer:

Follow the rules mentioned below while rationalizing a denominator:

  1. First, we have to check if all the given radicals or surds are in simplified form. 
  2. Then, find a suitable radical. Now multiply both the numerator and denominator with the radical such that it will remove the radicals in the denominator.

Question 4: What are conjugates in math?

Answer:

In mathematics, a number’s conjugate is a number that, when multiplied or added to the given number, yields a rational number.

For example,

  • The conjugate of a surd (a+√b) is (a−√b).
  • The conjugate of a complex number (a + ib) is (a − ib).

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