Class 9 Maths Chapter 1 Real Numbers Notes – Exercise 1.2 Solved (Free PDF Download)

Real Numbers is the first chapter of Class 9 Mathematics, and Exercise 1.2 is where the chapter becomes practical. After learning what rational and irrational numbers are in Exercise 1.1, this exercise teaches you to work with roots and powers — the skills that return in almost every later chapter and in board exams.

These notes cover Exercise 1.2 completely, in handwritten style and in very simple words. Every question is solved step by step, and each type of question is first explained with one small idea, so slow learners can follow the method without getting stuck.

Read This First: Roots and Powers Are Easier Than They Look

Let us be honest with each other. When most students open Exercise 1.2, the questions look frightening — big fractions, roots in strange places, powers with letters in them. Many students decide in the first five minutes that “this exercise is not for me”, and they copy the solutions without understanding a single line.

Here is the truth nobody tells you: this whole exercise runs on only three small tools. The first tool removes a root from the bottom of a fraction, using a twin called the conjugate. The second tool is a set of five power rules — flip for a negative power, root for a fraction power, and three more of the same size. The third tool is one friendly pair, x and 1/x, which build answer after answer once you find them. That is all. There is no fourth tool hiding anywhere in this exercise.

So do not try to memorise twenty-five answers. Learn the three tools, slowly, one at a time, and every question in Exercise 1.2 becomes a question you have already solved in a slightly different dress. These notes are arranged exactly in that order — tool first, questions after — so that by the time you reach Question 5, the “final boss” of the exercise, it will feel like a repeat, not a surprise.

Notes Details

Class	9
Subject	Mathematics
Chapter	1 – Real Numbers
Exercise	1.2 (complete)
Format	PDF (handwritten notes)
Pages	15
Language	Simple English
Price	Free

What Is Included in These Notes?

1. Rationalize the Denominator (Question 1)

All six parts of Question 1, solved with one repeatable recipe: write the conjugate, multiply the top and the bottom, and watch the root disappear from the denominator. Includes the harder parts where the numerator also has a root, such as (6 − 4√2)/(6 + 4√2) = 17 − 12√2.

2. Simplify the Powers (Question 2)

All nine parts of Question 2 — negative powers, fraction powers, roots of fractions with x, y and z, and the famous same-base game with letters n and x. Every big number is first changed into its small hidden base, like 81 = 3⁴ and 25 = 5².

3. The Special Value x = 3 + √8 (Question 3)

All six parts built as one chain. First the golden step — finding that 1/x = 3 − √8 — and then the sum, the difference, the squares and the fourth powers, each answer growing out of the previous one.

4. Find p and q (Question 4)

The complete method for writing (8 − 3√2)/(4 + 3√2) in the form p + q√2: rationalize first, then compare the two sides. The answer, p = −25 and q = 18, includes the negative-denominator trap explained slowly.

5. Final Simplifications (Question 5)

All four parts of Question 5, including the heavy fraction that simplifies to 3375/512 and the pattern-spotting question that ends in a + b². The notes finish with a one-page revision sheet and four try-yourself questions with answers.

Exercise 1.2 Detailed Walkthrough: Every Question Explained

Question 1 — Rationalize the denominator (6 parts). Rationalizing means removing the root from the bottom of a fraction, because a root in the denominator is considered unfinished in mathematics. The recipe never changes: look at the denominator, write its conjugate — the same expression with the middle sign flipped — and multiply both the numerator and the denominator by it. In part (i), 13/(4 + √3) is multiplied by (4 − √3)/(4 − √3); the denominator becomes 16 − 3 = 13, which cancels with the 13 above, leaving the clean answer 4 − √3. When the denominator is a single root, as in parts (ii) and (iii), you simply multiply top and bottom by that root itself: (√2 + √5)/√3 becomes (√6 + √15)/3, and (√2 − 1)/√5 becomes (√10 − √5)/5. Parts (iv) and (v) add one difficulty — the numerator also contains a root, so you must expand a square carefully using (a − b)² = a² + b² − 2ab. That is how (6 − 4√2)/(6 + 4√2) becomes (68 − 48√2)/4 = 17 − 12√2, and (√3 − √2)/(√3 + √2) becomes 5 − 2√6 over a denominator of 1. Part (vi), 4√3/(√7 + √5), finishes the set: multiply by (√7 − √5), the denominator becomes 7 − 5 = 2, and the answer is 2√3(√7 − √5).

Question 2 — Simplify (9 parts). This question practises the laws of exponents, and the notes begin every part the same way: change each big number into a small base. Part (i), (81/16)^(−3/4), starts by flipping the fraction because the power is negative, giving (16/81)^(3/4) = (2/3)³ = 8/27. Part (ii) is pure counting: after flipping (3/4)^(−2) into (4/3)², the 4s collect to 4¹ and the 3s to 3¹, so the answer is simply 12. Part (iii) shows that a decimal can hide a fraction: 0.027 is 27/1000, so (0.027)^(−1/3) becomes the cube root of 1000/27, which is 10/3. Part (iv) works under a seventh root — divide the powers of y and z inside first, then divide every power by 7, giving x²yz⁴. Parts (v), (vi), (viii) and (ix) all use letters, and all four surrender to the same idea: write everything in one base, take the common power out as a factor, and cancel it. That produces 1/6 in part (v), 9/2 in part (vi), 3⁵ = 243 in part (viii), and 95/5 = 19 in part (ix). Part (vii) is a neat division, (64)^(−2/3) ÷ (9)^(−3/2) = (1/16) ÷ (1/27) = 27/16.

Question 3 — If x = 3 + √8 (6 parts). Everything in this question grows from one golden step: rationalizing 1/x. Since (3 + √8)(3 − √8) = 9 − 8 = 1, the reciprocal is simply 1/x = 3 − √8, with no fraction left at all. Adding the pair, the roots cancel and x + 1/x = 6; subtracting them, the 3s cancel and x − 1/x = 2√8. From these two answers the rest is machinery: squaring the sum gives x² + 1/x² = 36 − 2 = 34; multiplying sum and difference gives x² − 1/x² = 6 × 2√8 = 12√8; squaring 34 gives x⁴ + 1/x⁴ = 1156 − 2 = 1154; and squaring the difference gives (x − 1/x)² = 4 × 8 = 32. The notes draw this as a chain so students see that no part starts from zero.

Question 4 — Find p and q. Here (8 − 3√2)/(4 + 3√2) must be written as p + q√2. Rationalizing with the conjugate (4 − 3√2) gives the numerator 50 − 36√2 and the denominator 16 − 18 = −2 — and that negative denominator is the trap of this question. Dividing carefully, the fraction equals −25 + 18√2. Comparing the plain part and the √2 part with p + q√2 gives p = −25 and q = 18.

Question 5 — Simplify (4 parts). The final question combines everything. Part (i) becomes (5³ × 3³)/(2⁵ × 2⁴) once each number is written in its small base, ending at 3375/512. Part (ii) cancels the common 3^(2x) above and below and leaves 54/81 = 2/3. Part (iii) works inside the square root first — (216)^(2/3) = 36, (25)^(1/2) = 5 and (0.04)^(−3/2) = 125 — so the root contains 36/25 and the answer is 6/5. Part (iv) rewards pattern-spotting: the expression is exactly A³ + B³ in the form (A + B)(A² − AB + B²) with A = a^(1/3) and B = b^(2/3), so it collapses to a + b².

Why These Notes Help Slow Learners

Most published solutions are written for students who are already fast. They jump three steps in one line, they never say why a step is taken, and they leave the slow student more confused than before. These notes are written the opposite way. Only three tools are introduced for the whole exercise, and every question is labelled by the tool it uses. Steps are kept to one rule per line. Traps are named before you fall into them — multiplying only the denominator, forgetting that a denominator can come out negative, or trying to expand a square in your head. If you have ever felt that mathematics is a race you keep losing, this exercise, done this way, is a good place to win some confidence back.

How to Use These Notes

Read the three tools on the first page until you can say them aloud. Then attempt Question 1 yourself and check each part against the notes. Move to Question 2 the same way — one part read, one part tried alone. For Question 3, always write the golden step (1/x = 3 − √8) at the top of your page before starting; every part uses it. Finish with the one-page revision sheet the night before a test, and try the four practice questions without looking at the answers.

Download Class 9 Maths Real Numbers Notes PDF

The complete Exercise 1.2 notes are available as a free PDF — the same handwritten pages described above, ready to read on your phone, print for your notebook, or share with your class fellows. No sign-up, no payment, just the download.

5 Common Mistakes Students Make in Exercise 1.2

1. Multiplying only the denominator. When rationalizing, whatever is multiplied below must also be multiplied above. Multiplying only the bottom changes the value of the fraction and loses every mark.

2. Forgetting the middle term of a square. (6 − 4√2)² is not 36 + 32. The middle term −2ab is where −48√2 comes from, and dropping it is the single most common error in Question 1.

3. Reading a negative power as a negative answer. A negative power means “flip”, not “become negative”. (81/16)^(−3/4) is the positive number 8/27, and (0.027)^(−1/3) is the positive 10/3.

4. Ignoring a negative denominator. In Question 4 the denominator after rationalizing is −2, not 2. Students who miss the minus sign report p = 25 and q = −18 — exactly upside down.

5. Starting every part of Question 3 from zero. Parts (iii) to (vi) are built from parts (i) and (ii). Students who recompute from scratch waste time and make fresh errors; students who use the chain finish quickly and correctly.

Frequently Asked Questions

Are these Class 9 Maths notes free?

Yes. The Exercise 1.2 PDF is completely free to download, print and share. Everything on All Free Assets is free.

Which exercise is covered in these notes?

Exercise 1.2 of Chapter 1 (Real Numbers) from the Class 9 Mathematics textbook — all five questions, including every part of Questions 1, 2, 3 and 5.

Are these notes suitable for slow learners?

Yes, they are written especially for slow learners: very simple English, one rule per line, each question labelled by the tool it uses, and a one-page revision sheet at the end.

Which syllabus do these notes follow?

They follow the PCTB (Punjab Curriculum and Textbook Board) Class 9 Mathematics book, Chapter 1 — Real Numbers, and they are equally useful for other BISE boards teaching the same chapter.

I am very weak in maths. Where should I start?

Start with Exercise 1.1 if you have not done it yet, then read the three tools on the first page of these notes and do Question 1 slowly. Do not touch Question 5 until Questions 1 and 2 feel easy — they use exactly the same tools.

Can I print these notes or share them with my class fellows?

Yes. The PDF prints cleanly on A4 paper, and you may share it freely with your class fellows and study groups.

Conclusion

Exercise 1.2 looks big from the outside, but inside it is only three tools used again and again: the conjugate, the five power rules, and the best-friends pair x and 1/x. Learn those slowly, question by question, and rationalizing denominators, simplifying powers and handling x = 3 + √8 will stop feeling like magic tricks and start feeling like routine. Download the free PDF, keep it with your textbook, and continue to the next exercise when you are ready. For the full chapter list, see our Free Class 9 Notes and Free Math Notes pages — and everything here on AllFreeAssets.com is free, always.