Exercise 1.5 in Class 9 Math Number System focuses on applying the laws of exponents to real numbers. In this exercise, students practice simplifying expressions using exponent rules, helping build a strong foundation for algebraic operations.
Class 9 Math Number System Ex 1.5 – Textbook
Question 1.
Q: Classify the following numbers as rational or irrational:
(i) 2 – √5
- √5 is an irrational number.
- Subtracting an irrational number from a rational number (2 – √5) gives an irrational number.
Answer: Irrational
(ii) (3 + √23) – √23
- Simplify the expression:
(3 + √23) – √23 = 3 - 3 is a rational number.
Answer: Rational
(iii) (2√7) / (7√7)
- Cancel √7 from numerator and denominator:
(2√7) / (7√7) = 2 / 7 - 2/7 is a rational number.
Answer: Rational
(iv) 1 / √2
- √2 is irrational.
- The reciprocal of an irrational number is also irrational.
Answer: Irrational
(v) 2π
- π is irrational.
- Multiplying an irrational number (π) by a rational number (2) gives an irrational number.
Answer: Irrational
2. (i) Simplify:
(i) (3 + √3)(2 + √2)
= 3×2 + 3×√2 + √3×2 + √3×√2
= 6 + 3√2 + 2√3 + √6
(ii) (3 + √3)(3 – √3)
= 3² – (√3)²
= 9 – 3
= 6
(iii) (√5 + √2)²
= (√5)² + 2×√5×√2 + (√2)²
= 5 + 2√10 + 2
= 7 + 2√10
(iv) (√5 – √2)(√5 + √2)
= (√5)² – (√2)²
= 5 – 2
= 3
Question 3:
Recall, π is defined as the ratio of the circumference (say c) of a circle to its diameter (say d). That is,
π = c/d. This seems to contradict the fact that π is irrational. How will you resolve this contradiction?
Answer:
Actually, c/d = 22/7 is only an approximate value of π and also a non-terminating decimal.
The exact value of π is irrational, meaning it cannot be expressed as a ratio of two integers. Its decimal expansion is non-terminating and non-repeating.
Hence, 22/7 is just a rational approximation of π. So, there is no contradiction.
Question 4:
Represent √9.3 on the number line.
Answer:
To represent √9.3 on the number line:
- Draw a line segment AB = 9.3 units.
- Extend the line 1 unit further to point C, so AC = 10.3 units.
- Find the midpoint M of AC.
- With M as center and radius MA, draw a semicircle.
- Draw a perpendicular from point B to meet the semicircle at point D.
- Length BD = √9.3.
- With B as center and BD as radius, draw an arc on the number line — this arc represents √9.3.
This geometric method helps us represent √9.3 on the number line
(5) Rationalise the denominator of:
(i)1 / √7
Multiply numerator and denominator by √7:
1 / √7 × √7 / √7 = √7 / 7
Answer: √7 / 7
(ii) Rationalise the denominator of:
1 / (√7 − √6)
Multiply numerator and denominator by (√7 + √6):
1 / (√7 − √6) × (√7 + √6) / (√7 + √6)
= (√7 + √6) / [(√7)² − (√6)²]
= (√7 + √6) / (7 − 6)
= √7 + √6
Answer: √7 + √6
(iii) Rationalise the denominator of:
1 / (√5 + √2)
Multiply numerator and denominator by (√5 − √2):
1 / (√5 + √2) × (√5 − √2) / (√5 − √2)
= (√5 − √2) / [(√5)² − (√2)²]
= (√5 − √2) / (5 − 2)
= (√5 − √2) / 3
Answer: (√5 − √2) / 3
(iv) Rationalise the denominator of:
1 / (√7 − 2)
Multiply numerator and denominator by (√7 + 2):
1 / (√7 − 2) × (√7 + 2) / (√7 + 2)
= (√7 + 2) / [(√7)² − 2²]
= (√7 + 2) / (7 − 4)
= (√7 + 2) / 3
Answer: (√7 + 2) / 3
The questions and solutions provided on this page are based on the Class 9 Math Number System Ex 1.5. For detailed study and official content, you can refer to the NCERT textbook available on the official NCERT website.
Class-wise Solutions
Class 12:
Class 12 Physics – NCERT Solutions
Class 12 Chemistry – NCERT Solutions
Class 11:
- Class 11 Physics – NCERT Solutions
- Class 11 Chemistry – NCERT Solutions
- Class 11 Biology – NCERT Solutions
- Class 11 Math – NCERT Solutions
Class 10:
Class 9:
Class 8:
Class 7:
Class 6:
Subject-wise Solutions
Physics:
Chemistry:
Biology:
Math:
- Class 11 Math – NCERT Solutions
- Class 10 Math – NCERT Solutions
- Class 9 Math – NCERT Solutions
- Class 8 Math – NCERT Solutions
Science:
- Class 10 Science – NCERT Solutions
- Class 9 Science – NCERT Solutions
- Class 8 Science – Oxford Solutions
- Class 7 Science – Oxford Solutions
- Class 6 Science – Oxford Solutions
NEET BIOLOGY
- Evolution
- Breathing and Exchange of Gases
- Anatomy of Flowering Plants
- Body Fluids and Circulation
- Human Health and Disease
- Microbes in Human Welfare
- Cell Cycle and Cell Division
- Biotechnology and Its Applications
- Biodiversity and Conservation
- Morphology of Flowering Plants
For the official Class 8 Mathematics Solutions, you can visit:
- NCERT Textbooks (for Class 8):
In this set of questions from Class 9 Math Number System Ex 1.5, we practiced rationalizing denominators, an important concept when working with irrational numbers. Understanding how to simplify such expressions is essential, and Class 9 Math Number System Ex 1.5 provides a strong foundation for it. Each question in Class 9 Math Number System Ex 1.5 reinforces the method of multiplying by conjugates or suitable surds to make the denominator rational.