Rationalize the denominator of √5/(√6 + √2)

The system that represents numbers is known as the number system, where a number is a mathematical value that helps in counting or measuring, or labeling objects, and also performing different mathematical calculations. It is the mathematical notation that is used to consistently represent the numbers in a given set using digits or other symbols. It represents the arithmetic and algebraic structure of the numbers and gives each number a distinct representation. It also enables us to perform mathematical operations like addition, subtraction, and division. A number system is classified into various types, such as a decimal number system, a binary number system, an octal number system, and a hexadecimal number system. 

Radicals

A radical is an expression that has a root, such as a square root, a cube root, etc. The root is considered to be a square root if the index of a radical expression is not mentioned. For instance, the “4th root of (x + y)” can be symbolically written as shown in the figure given below. Here, “4” is the index or degree, “(x+ y)” is the radicand, and “(⁴√)’ is the radical symbol. If an irrational number, which is the radical form, cannot be simplified further, then the number is called a surd. For example, √5 is a surd as it cannot be simplified further.

Rationalization

Rationalization is a process that is used to make the denominator of an algebraic fraction free from an irrational number. In rationalization, we eliminate a radical or an imaginary number from the denominator such that it only contains a rational number. In math, conjugates are a pair of binomials that have identical terms but share opposite signs between their two terms. For example, (a + b) is a conjugate of (a − b) and vice versa. 

Rationalizing a Monomial Radical

A monomial radical is a polynomial radical that has only one term. √3, √2a, ⁴√4x, etc are examples of monomial radicals. Based on the degree of the polynomial, different steps have to be followed to rationalize a surd or radical in the denominator. Let us consider a fraction with a radical in the denominator, i.e., a/√b. Now to rationalize the denominator, multiply both numerator and denominator with √b. 

Example: Rationalize: 7 / ⁴√x.

Solution:

Given: 7 / ⁴√x = 7 / (x)^1/4

Now, multiply both numerator and denominator with (x)^3/4.

= 7 / (x)1/4 × (x)^3/4 / (x)^3/4

= 7(x)^3/4/x^(1/4+3/4)

= 7(x)^3/4 / x = 7(x³)^1/4 / x 

=> 7 / ⁴√x  = 7 ⁴√x³ / x

Rationalizing a Binomial Radical

If the denominator of a radical expression is in the form of (a+√b) or (a + i √b), then multiply both the numerator and denominator with the conjugate of the expression, i.e., (a−√b) or (a − i √b).

Example: Rationalize: 1/(3 − √5).

Solution:

Given: 1/(3 − √5)

To rationalize the denominator, multiply and divide the given term with (3 + √5).

= 1/(3 − √5) × (3 + √5)/(3 + √5)

Since, (a + b)(a – b) = a² – b²

(3 − √5)×(3 + √5) = (3)² − (√5)² = 9 − 5 = 4

=> 1/(3 − √5) = (3 + √5)/4.

How to Rationalize the Denominator?

To rationalize the denominator of a fraction, follow the steps mentioned below.

Step 1: Multiply both the denominator and numerator of the given fraction by a suitable radical that will remove the radicals in the denominator.

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

Step 3: Simplify the fraction further if necessary.

Rationalize the denominator of √5/(√6 + √2)

Solution:

Given, √5/(√6 + √2)

To rationalize the denominator, multiply and divide the given term with (√6 – √2).

= √5/(√6 + √2) × (√6 – √2)/(√6 – √2)

Since, (a + b)(a – b) = a² – b²

(√6 + √2)×(√6 – √2) = (√6)² – (√2)² 

= 6 – 2 = 4

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

= [√(5×6) – √(5×2)]/4

= (√30 – √10)/4

Hence, √5/(√6 + √2) = (√30 – √10)/4.

Practise Problems based on Rationalization

Problem 1: Rationalize: 1/(√2 − √7). 

Solution:

Given: 1/(√2 − √7).

To rationalize the denominator, multiply and divide the given term with (√2 + √7).

= 1/(√2 − √7) × (√2 + √7)/(√2 + √7)

Since, (a + b)(a – b) = a² – b²

(√2 − √7)×(√2 + √7) = (√2)² − (√7)² 

= 2 − 7 = −5

1/(√2 – √7) × (√2 + √7)/(√2 + √7)

= (√2 + √7)/(-5)

= −(√2 + √7)/5

Hence, 1/(√2 – √7) = −(√2 + √7)/5

Problem 2: Rationalize: 1/(4 − √6).

Solution:

Given: 1/(4 − √6).

To rationalize the denominator, multiply and divide the given term with (4 + √6).

= 1/(4 − √6) × (4 + √6)/(4 + √6)

Since, (a + b)(a – b) = a² – b²

(4 − √6)×(4 + √6) = (4)2 − (√6)2

= 16 −6 = 10

1/(4 − √6) × (4 + √6)/(4 + √6) = (4 + √6)/10

Hence, 1/(4 − √6) = (4 + √6)/10.

Problem 3: Rationalize: a) 3/√5   b) 5 / ³√4.

Solution:

• 3/√5

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

= 3/√5 × √5/√5

= 3√5/5

Hence, 3/√5 = 3√5/5.

• 5 / ³√4 = 5/(4)^1/3

To rationalize the denominator, multiply and divide the given term with 4^2/3.

= 5/(4)^1/3 × 4^2/3/4^2/3

= 5 × 4^2/3  /4^{(1/3 + 2/3)}

= 5 × (4²)^1/3/4

= 5/4 × 4^2/3

Thus, 5/³√4 =  5/4 × 4^2/3

Problem 4: Rationalize: −3/(1 + √2).

Solution:

Given: −3/(1 + √2)

To rationalize the denominator, multiply and divide the given term with (1 − √2).

= −3/(1 + √2) × (1 − √2)/(1 − √2)

Since, (a + b)(a – b) = a² – b²

(1 + √2)×(1 − √2) = (1)2 − (√2)2

= 1 − 2 = − 1

−3/(1 + √2) × (1 − √2)/(1 − √2) = −3(1 − √2)/(−1)

= 3(1 − √2)

Hence, −3/(1 + √2) = 3(1 − √2).

Problem 5: Rationalise: −6/√3

Solution:

 −6/√3

To rationalize the denominator, multiply and divide the given term with √3.

= −6/√3 × √3/√3

= −6√3/3

Hence,

−6/√3 = −6√3/3.

Frequently Asked Questions on Rationalization of Denominator

Question 1: How to rationalize the denominator that contains a square root?
Answer:

 To rationalise the denominator that contains a square root, we multiply and divide the given rational number with the same square root value. In this way, the denominator changes to a rational number.

Question 2: What is meant by rationalization of denominator?
Answer:

Rationalisation of denominator signifies to remove any radical term or surds from the denominator and expressing the rational number in a simplified form.

Question 3: What value cannot be used in the denominator?
Answer: 

Denominator of any rational number cannot be zero, as zero in denominator is undefined number.

Question 4: How do we rationalize the denominator which contains two terms?
Answer:

For rationalising the denominator that contains two terms, firstly we find the conjugate pair of the denominator and then multiply and divide the fraction by the sme conjugate pair of the denominator.