Conjugate in Math

Conjugate in Math

Hello kids! Do you know what conjugate means?

The word conjugate means a couple of objects that have been linked together. Except for one pair of characteristics that are actually opposed to each other, these two items are the same.

Let us learn about this more in detail.

Conjugate in math

Lesson Plan

What Is Meant by Conjugate in Math? 

Conjugate in math means to write the negative of the second term.

By flipping the sign between two terms in a binomial, a conjugate in math is formed.

The conjugate of \(a+b\) can be written as \(a-b\).


How to Conjugate Binomials?

The conjugate of binomials can be found out by flipping the sign between two terms.

For example the conjugate of \(m+n\) is \(m-n\). In other words, it can be also said as \(m+n\) is conjugate of \(m-n\).

The conjugate of \(5x + 2 \) is \(5x - 2 \)

Conjugate in math

Look at the table given below of conjugate in math which shows a binomial and its conjugate.

Binomial Conjugate of Binomial
\(3 - \sqrt 2 \) \(3 + \sqrt 2 \)
\(5 + \sqrt 7 \) \(5 - \sqrt 7 \)
\(\sqrt 7  - \sqrt 2 \) \(\sqrt 7  + \sqrt 2 \)
\(\sqrt 5  -  11 \) \(\sqrt 5  +  11 \)
\(\sqrt 5  + \frac{1}{3}\) \(\sqrt 5  - \frac{1}{3}\)

How to Rationalize the Denominator Using Conjugates?

To rationalize the denominator using conjugate in math, there are certain steps to be followed.

Let us understand this by taking one example.

Example

Rationalize the denominator  \(\frac{1}{{5 - \sqrt 2 }}\)

Solution

Step 1: Find out the conjugate of the number which is to be rationalized. In our case that is \(5 + \sqrt 2 \)

Step 2: Now multiply the conjugate, i.e.,  \(5 + \sqrt 2 \) to both numerator and denominator. 

\[\begin{align}
  &= (\frac{1}{{5 - \sqrt 2 }}) \times (\frac{{5 + \sqrt 2 }}{{5 + \sqrt 2 }}) \\[0.2cm] 
  &= \frac{{5 + \sqrt 2 }}{{(5 - \sqrt 2 )(5 + \sqrt 2 )}} \\[0.2cm]  
  &= \frac{{5 + \sqrt 2 }}{{(5)^2 - (\sqrt 2 )^2}} \\[0.2cm]  
 &= \frac{{5 + \sqrt 2 }}{{25 - 2}} \\[0.2cm]  
 &= \frac{{5 + \sqrt 2 }}{{23}} \\
 \end{align}\]

So this is how we can rationalize denominator using conjugate in math.

 
important notes to remember
Important Notes
  1. In math, the conjugate implies writing the negative of the second term.

  2. Binomial conjugate can be explored by flipping the sign between two terms.

  3. While solving for rationalizing the denominator using conjugates, just make a negative of the second term and multiply and divide it by the term.
    For \(\frac{1}{{a + b}}\) the conjugate is \(a-b\) so, multiply and divide by it.

Solved Examples

Example 1

 

 

Rationalize \(\frac{4}{{\sqrt 7  + \sqrt 3 }}\)

Solution

\[\begin{align}
  &= \frac{4}{{\sqrt 7  + \sqrt 3 }} \times \frac{{\sqrt 7  - \sqrt 3 }}{{\sqrt 7  - \sqrt 3 }} \\[0.2cm] 
  &= \frac{{4(\sqrt 7  - \sqrt 3 )}}{{(\sqrt 7 )^2 - (\sqrt 3 )^2}} \\[0.2cm]  
  &= \frac{{4(\sqrt 7  - \sqrt 3 )}}{{7 - 3}} \\[0.2cm]  
  &= \frac{{4(\sqrt 7  - \sqrt 3 )}}{4} \\[0.2cm]  
  &= \sqrt 7  - \sqrt 3  \\[0.2cm]  
 \end{align}\]

\(\therefore \text {The answer is} \sqrt 7  - \sqrt 3 \)
Example 2

 

 

Rationalize \(\frac{{5 + 3\sqrt 2 }}{{5 - 3\sqrt 2 }}\)

Solution

\[\begin{align}
  &= \frac{{(5 + 3\sqrt 2 )}}{{(5 - 3\sqrt 2 )}} \times \frac{{(5 + 3\sqrt 2 )}}{{(5 + 3\sqrt 2 )}} \\[0.2cm] 
  &= \frac{{(5 + 3\sqrt 2 )2}}{{(5)^2 - (3\sqrt 2 )^2}} \\[0.2cm]  
  &= \frac{{(5)^2 + 2(5)(3\sqrt 2 ) + (3\sqrt 2 )^2}}{{(25) - (18)}} \\[0.2cm]  
  &= \frac{{25 + 30\sqrt 2  + 18}}{7} \\[0.2cm]  
  &= \frac{{43 + 30\sqrt 2 }}{7} \\[0.2cm]  
 \end{align}\]

\(\therefore \text {The answer is} \frac{{43 + 30\sqrt 2 }}{7} \)
Example 3

 

 

Find the value of  \(3 + \frac{1}{{3 + \sqrt 3 }}\)

Solution

\[\begin{align}
 3 + \frac{1}{{3 + \sqrt 3 }} \\[0.2cm] 
  = 3 + \frac{1}{{3 + \sqrt 3 }} \times \frac{{3 - \sqrt 3 }}{{3 - \sqrt 3 }} \\[0.2cm] 
  = 3 + \frac{{3 - \sqrt 3 }}{{(3 + \sqrt 3 )(3 - \sqrt 3 )}} \\[0.2cm]  
  = 3 + \frac{{3 - \sqrt 3 }}{{(3)^2 - (\sqrt 3 )^2}} \\[0.2cm]
  = 3 + \frac{{3 - \sqrt 3 }}{{9 - 3}} \\[0.2cm]  
  = 3 + \frac{{3 - \sqrt 3 }}{6} \\[0.2cm]  
  = \frac{{18 + 3 - \sqrt 3 }}{6} \\[0.2cm]  
  = \frac{{21 - \sqrt 3 }}{6} \\[0.2cm]
 \end{align}\]

\(\therefore \text {The answer is} \frac{{21 - \sqrt 3 }}{6} \)
Example 4

 

 

Find the value of a and b in \(\frac{{3 + \sqrt 7 }}{{3 - \sqrt 7 }} = a + b\sqrt 7 \)

Solution

\( \frac{{3 + \sqrt 7 }}{{3 - \sqrt 7 }} = a + b\sqrt 7\)
\[\begin{align}
 \text{LHS} &= \frac{{3 + \sqrt 7 }}{{3 - \sqrt 7 }} \times \frac{{3 + \sqrt 7 }}{{3 + \sqrt 7 }} \\ 
  &= \frac{{(3 + \sqrt 7 )2}}{{(3)^2 - (\sqrt 7 )^2}} \\ 
  &= \frac{{(3)^2 + 2(3)(\sqrt 7 ) + (\sqrt 7 )^2}}{{9 - 7}} \\ 
  &= \frac{{9 + 6\sqrt 7  + 7}}{2} \\ 
  &= \frac{{16 + 6\sqrt 7 }}{2} \\ 
  &= \frac{{2(8 + 3\sqrt 7 )}}{2} \\ 
  &= 8 + 3\sqrt 7  \\ 
 \end{align}\]
\[\begin{align}
 8 + 3\sqrt 7  = a + b\sqrt 7  \\[0.2cm] 
 \therefore a = 8\ and\  b = 3 \\ 
 \end{align}\]

\(\therefore \text {The value of }a = 8\ and\  b = 3\)
Example 5

 

 

If \(\ x = 2 + \sqrt 3 \) find the value of \( x^2 + \frac{1}{{x^2}}\)

Solution

\[(x + \frac{1}{x})^2 = x^2 + \frac{1}{{x^2}} + 2.........(1)\]

So we need \(\frac{1}{x}\) to get the value of \(x^2 + \frac{1}{{x^2}}\)

\[\begin{align}
 \therefore \frac{1}{x} &= \frac{1}{{2 + \sqrt 3 }} \\[0.2cm] 
  &= \frac{1}{{2 + \sqrt 3 }} \times \frac{{2 - \sqrt 3 }}{{2 - \sqrt 3 }} \\[0.2cm]  
  &= \frac{{2 - \sqrt 3 }}{{(2)^2 - (\sqrt 3 )^2}} \\[0.2cm]  
  &= \frac{{2 - \sqrt 3 }}{{4 - 3}} \\[0.2cm]
 \frac{1}{x} &= 2 - \sqrt 3  \\
 \end{align}\]
Substitute both \(x\) & \(\frac{1}{x}\) in statement number 1

\[\begin{align}
 [(2 + \sqrt 3 ) + (2 - \sqrt 3 )]^2 &= x^2 + \frac{1}{{x^2}} + 2 \\ 
 (4)^2 &= x^2 + \frac{1}{{x^2}} + 2 \\ 
 16 &= x^2 + \frac{1}{{x^2}} + 2 \\ 
 16 - 2 &= x^2 + \frac{1}{{x^2}} \\ 
 \therefore\ x^2 + \frac{1}{{x^2}} &= 14 \\
 \end{align}\]

\(\therefore  x^2 + \frac{1}{{x^2}} = 14\) 
 
Challenge your math skills
Challenging Questions
  • Rationalize \(\frac{1}{{\sqrt 6  + \sqrt 5  - \sqrt {11} }}\)
  • If \(a = \frac{{\sqrt 3  - \sqrt 2 }}{{\sqrt 3  + \sqrt 2 }}\) and \(b = \frac{{\sqrt 3  + \sqrt 2 }}{{\sqrt 3  - \sqrt 2 }}\), find the value of \(a^2+b^2-5ab\).

Interactive Questions

Here are a few activities for you to practice. 

Select/Type your answer and click the "Check Answer" button to see the result

 
 
 
 
 
 

Let's Summarize

The mini-lesson targeted the fascinating concept of Conjugate in Math. The math journey around Conjugate in Math starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.

About Cuemath

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Frequently Asked Questions (FAQs)

1. What is the conjugate of 1?

The conjugate of 1 is 1

2. What is a conjugate pair?

A conjugate pair means a binomial which has a second term negative. We only have to rewrite it and alter the sign of the second term to create a conjugate of a binomial. 

3. What is the conjugate of 5?

It doesn't matter whether we express 5 as an irrational or imaginary number. The conjugate of 5 is, thus, 5

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