Find the best tutors and institutes for Class 10 Tuition
Q5(iii):
Rationalise the denominators of the following:
(iii) $\frac{1}{\sqrt{5} + \sqrt{2}}$
Solution :
Initial Setup & Objective
We are given the following fractional expression containing irrational terms in the denominator:
$ \frac{1}{\sqrt{5} + \sqrt{2}} $
The mathematical objective is to rationalise the denominator. Rationalisation is the process of eliminating irrational numbers (such as surds or square roots) from the denominator of an algebraic fraction. This is achieved by multiplying the numerator and the denominator by a carefully chosen multiplier known as the conjugate.
Step 1: Identifying the Conjugate (Rationalising Factor)
To eliminate the square roots in a binomial denominator of the form $\sqrt{a} + \sqrt{b}$, we utilize its conjugate pair, which is $\sqrt{a} - \sqrt{b}$. [Per the fundamental properties of surds, multiplying a binomial by its conjugate leverages the difference of squares identity to yield a rational integer].
For our specific denominator, $\sqrt{5} + \sqrt{2}$, the conjugate is:
$ \sqrt{5} - \sqrt{2} $
| Original Binomial Denominator | Conjugate (Rationalising Factor) | Algebraic Product (Rational Result) |
|---|---|---|
| $\sqrt{a} + \sqrt{b}$ | $\sqrt{a} - \sqrt{b}$ | $(\sqrt{a})^2 - (\sqrt{b})^2 = a - b$ |
| $\sqrt{a} - \sqrt{b}$ | $\sqrt{a} + \sqrt{b}$ | $(\sqrt{a})^2 - (\sqrt{b})^2 = a - b$ |
| $a + \sqrt{b}$ | $a - \sqrt{b}$ | $a^2 - (\sqrt{b})^2 = a^2 - b$ |
Step 2: Multiplying Numerator and Denominator
To maintain the equivalence of the fraction [Per the Multiplicative Identity Property, multiplying by $1$ does not change the value], we multiply both the numerator and the denominator by the conjugate $\sqrt{5} - \sqrt{2}$:
$ \frac{1}{\sqrt{5} + \sqrt{2}} \times \frac{\sqrt{5} - \sqrt{2}}{\sqrt{5} - \sqrt{2}} $
Step 3: Applying the Difference of Squares Identity
We now expand the numerator and the denominator. The numerator is simply multiplied by $1$:
$ \text{Numerator} = 1 \times (\sqrt{5} - \sqrt{2}) = \sqrt{5} - \sqrt{2} $
For the denominator, we apply the algebraic identity for the difference of squares: $(x + y)(x - y) = x^2 - y^2$.
Let $x = \sqrt{5}$ and $y = \sqrt{2}$:
$ \text{Denominator} = (\sqrt{5} + \sqrt{2})(\sqrt{5} - \sqrt{2}) = (\sqrt{5})^2 - (\sqrt{2})^2 $
Step 4: Evaluating the Squares and Final Simplification
Evaluating the squares in the denominator [Per the definition of a square root, $(\sqrt{n})^2 = n$ for any non-negative real number $n$]:
- $(\sqrt{5})^2 = 5$
- $(\sqrt{2})^2 = 2$
Substitute these values back into the denominator's expression:
$ \text{Denominator} = 5 - 2 = 3 $
Now, recombine the evaluated numerator and the rationalised denominator to form the final simplified fraction:
$ \frac{\sqrt{5} - \sqrt{2}}{3} $
The denominator is now $3$, which is a rational integer. The rationalisation process is complete.
Final Solution: $ \frac{\sqrt{5} - \sqrt{2}}{3} $
More Questions from Class 9 Mathematics Number Systems EXERCISE 1.4
- Q1(i): Classify the following numbers as rational or irrational: (i) $2 - \sqrt{5}$
- Q1(ii): Classify the following numbers as rational or irrational: (ii) $(3 + \sqrt{23}) - \sqrt{23}$
- Q1(iii): Classify the following numbers as rational or irrational: (iii) $\frac{2\sqrt{7}}{7\sqrt{7}}$
- Q1(iv): Classify the following numbers as rational or irrational: (iv) $\frac{1}{\sqrt{2}}$
- Q1(v): Classify the following numbers as rational or irrational: (v) $2\pi$
- Q2(i): Simplify each of the following expressions: (i) $(3 + \sqrt{3})(2 + \sqrt{2})$
- Q2(ii): Simplify each of the following expressions: (ii) $(3 + \sqrt{3})(3 - \sqrt{3})$
- Q2(iii): Simplify each of the following expressions: (iii) $(\sqrt{5} + \sqrt{2})^2$
- Q2(iv): Simplify each of the following expressions: (iv) $(\sqrt{5} - \sqrt{2})(\sqrt{5} + \sqrt{2})$
- Q3: Recall, $\pi$ is defined as the ratio of the circumference (say $c$) of a circle to its diameter (say $d$). That is, $\pi = \frac{c}{d}$. This seems to contradict the fact that $\pi$ is irrational. How will you resolve this contradiction?
- Q4: Represent $\sqrt{9.3}$ on the number line.
- Q5(i): Rationalise the denominators of the following: (i) $\frac{1}{\sqrt{7}}$
- Q5(ii): Rationalise the denominators of the following: (ii) $\frac{1}{\sqrt{7} - \sqrt{6}}$
- Q5(iv): Rationalise the denominators of the following: (iv) $\frac{1}{\sqrt{7} - 2}$
CBSE Solutions for Class 9 Mathematics Number Systems
Chapters in CBSE - Class 9 Mathematics
Top Tutors who teach Number Systems
It was a great experience while taking the class tenth private tutions.
Fabulous experience learning with Indrajeet Sir, quickly made me grasp difficult concepts in preparation for my board examinations.
Your Hard Work. Their Future. Our Responsibility. At Fundamend Academy, we know that every penny you spend on tuition is a penny earned through sacrifice. You aren't just looking for "classes"—you are looking for a teacher who cares as much as you do. Why Fundamend Academy? We’ve removed the risks that usually waste your money and time: Money-Back Peace of Mind: If your child doesn't "click" with their teacher, we swap the tutor immediately at zero extra cost. Your fees will never go to waste. (1) Zero Hidden Costs: All study materials, board exam test series, and session recordings are included. No extra books to buy. (2) Total Honesty: No fake promises. You get a weekly report and a monthly call to discuss your child’s real progress. (3) The Comfort of Home Language: We teach in your child’s native tongue so they finally understand the concept instead of just memorizing it. (4) Total Honesty: No fake promises. You get a weekly report and a monthly call to discuss your child’s real progress. We don’t just teach; we take the burden off your shoulders so you can see your child succeed. BOOK A FREE DEMO See our teaching for yourself before you spend a single rupee.
I thank from my bottom of my heart to Girish Sir for identifying the gaps in my son, now he secured excellent marks in his boards. Thanks for all the tutors. I can't forget this. I strongly recommend my friends, relatives .
25 years of private teaching experience in maths and science at Mumbai western line.
In this competitive world education is the most important tool rather I say that it is sword by which one can win the competition.so I am in this field from last four years and teaching is my passion and even strength.My students are very happy with me in college as well as in tutions. Their results show it all. Students are always been encouraged to do studies one should not force them for the same.
I teach Mathematics to students from Class 7 to Class 10, with 3 years of teaching experience. My focus is on helping students build strong conceptual understanding and develop effective problem-solving skills to excel academically. I have completed my 2nd PUC (Pre-University Course), a Bachelor of Engineering (B.E.) from Visveswaraya Technological University in 2020, and a Bachelor of Education (B.Ed.) from Indira Gandhi National Open University in 2022. I have also completed a program with Cambridge Learning Resources, which has enhanced my teaching techniques and ability to engage students effectively. Over the years, I have guided students to improve their Mathematics understanding and performance, helping them gain confidence in handling both basic and complex problems. My teaching approach emphasizes clarity, practice, and logical reasoning to ensure steady improvement. I have also earned an RPA (Robotic Process Automation) certification in robotics, which reflects my technical skills and commitment to continuous learning. I often incorporate logical thinking and problem-solving strategies inspired by this training into my lessons. I prefer to conduct online classes, offering flexibility in timing and easy access for students. I teach both individual and small group sessions, allowing personalized attention and interactive learning experiences for every student. My classes are interactive, using motivational videos, personal digital pads, and other digital tools to make learning engaging and effective. These resources help students visualize concepts better and retain knowledge more efficiently. I provide custom-prepared notes to support student learning and revision. My ultimate goal is to make Mathematics understandable, enjoyable, and confidence-building, enabling every student to perform at their best and develop a strong foundation for future studies.
This guy is too good at his teaching. He explains the concepts very clear, gets to know about the students, gives his own ideas, own methods, and also gives good advices. He is excellent to consider. I just like him!, he is a great tutor and price is reasonably good.
Find more Tutor for Number Systems in your City
- Bangalore Mathematics Tutors
- Delhi Mathematics Tutors
- Chennai Mathematics Tutors
- Gurgaon Mathematics Tutors
- Noida Mathematics Tutors
- Hyderabad Mathematics Tutors
- Mumbai Mathematics Tutors
- Ghaziabad Mathematics Tutors
- Chandigarh Mathematics Tutors
- Pune Mathematics Tutors
- Jaipur Mathematics Tutors
- Surat Mathematics Tutors
Download free CBSE - Class 9 Mathematics Number Systems EXERCISE 1.4 worksheets
Download Now