Class 9 Math Number System Ex 1.5 NCERT Solution

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

Class10 Math NCERT Solutions|

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.

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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.