Prove the following number is irrational: 5 4

Let assume 5 4 is rational number. Let 5 4 = p q , where p and q are integers, q > 0 and p and q have no common factors Taking four power on both sides, we get 5 = p 4 q 4 ⇒ 5 q 4 = p 4 Therefore, 5 q 4 is divisible by 5 which dictates that p 4 is divisible by 5 . So, p is also divisible by 5 . Let p = 5 k , where k is an integer. Taking four power on both sides, we get p 4 = 625 k 4 ⇒ 5 q 4 = 625 k 4 ⇒ q 4 = 125 k 4 Therefore, 125 k 4 is divisible by 5 which dictates that q 4 is divisible by 5 . So, q is also divisible by 5 . Hence, p and q both are divisible by 5 . But, p and q are co-prime as per assumption so our assumption is wrong. Thus 5 4 is an irrational number.

Write the following numbers in ascending order: 3 2 , 2 3 , 15 , 4

Given numbers are 3 2 , 2 3 , 15 , 4 Write all the numbers as square roots under one radical. 3 2 = 9 × 2 = 18 2 3 = 4 × 3 = 12 15 = 15 4 = 16 Since 12 < 15 < 16 < 18 ⇒ 2 3 < 15 < 4 < 3 2 Hence, the given numbers in ascending order are, 2 3 , 15 , 4 , 3 2

Write the following numbers in ascending order: 3 2 , 2 8 , 4 , 50 , 4 3

Given numbers are 3 2 , 2 8 , 4 , 50 , 4 3 Write all the numbers as square roots under one radical, 3 2 = 9 × 2 = 18 2 8 = 4 × 8 = 32 4 = 16 50 = 50 4 3 = 16 × 3 = 48 Since 16 < 18 < 32 < 48 < 50 ⇒ 4 < 3 2 < 2 8 < 4 3 < 50 Hence , the given numbers in ascending orders are 4 , 3 2 , 2 , 8 , 4 3 , 50

Write the following real numbers in descending order: 9 2 , 3 2 5 , 4 3 , 3 6 5

Given numbers are 9 2 , 3 2 5 , 4 3 , 3 6 5 Write all the numbers as square roots under one radical. 9 2 = 81 2 = 40 . 5 3 2 5 = 9 4 × 5 = 45 4 = 11 . 25 4 3 = 16 × 3 = 48 3 6 5 = 9 × 6 5 = 54 5 = 10 . 8 Since 48 > 40 . 5 > 11 . 25 > 10 . 8 ⇒ 48 > 40 . 5 > 11 . 25 > 10 . 8 ⇒ 4 3 > 9 2 > 3 2 5 > 3 6 5 Hence, the given numbers in descending order are 4 3 , 9 2 , 3 2 5 , 3 6 5 .

Write the following real numbers in descending order: 5 3 , 7 3 2 , - 3 , 3 5 , 2 7

Given numbers are 5 3 , 7 3 2 , - 3 , 3 5 , 2 7 Write all the numbers as square roots under one radical. 5 3 = 25 3 = 8 . 33 7 3 2 = 49 9 × 2 = 98 9 = 10 . 89 - 3 = - 3 3 5 = 9 × 5 = 45 2 7 = 4 × 7 = 28 Since 45 > 28 > 10 . 89 > 8 . 33 ⇒ 45 > 28 > 10 . 89 > 8 . 33 > - 3 ⇒ 3 5 > 2 7 > 7 3 2 > 5 3 > - 3 Hence, the given numbers in descending order are 3 5 , 2 7 , 7 3 2 , 5 3 , - 3 .

Arrange the following numbers in ascending order: 2 3 , 3 , 5 6 .

Given numbers are 2 3 , 3 , 5 6 Write all the numbers under one radical. L.C.M of 2 , 3 and 6 is 6 2 3 = 2 1 3 = 2 2 1 6 = 4 1 6 3 = 3 1 2 = 3 3 1 6 = 27 1 6 5 6 = 5 1 6 = 5 1 6 Since 4 < 5 < 27 ⇒ 4 1 6 < 5 1 6 < 27 1 6 ⇒ 2 3 < 5 6 < 3 Hence, the given numbers in ascending order are 2 3 , 5 6 , 3 .

E M B I B E

Understanding ICSE Mathematics Class 9>Pure Arithmetic>Chapter 1 - Rational and Irrational Numbers>EXERCISE 1.4

M L Aggarwal Solutions for Exercise 4: EXERCISE 1.4

Author:M L Aggarwal

M L Aggarwal Mathematics Solutions for Exercise - M L Aggarwal Solutions for Exercise 4: EXERCISE 1.4

Attempt the free practice questions from Exercise 4: EXERCISE 1.4 with hints and solutions to strengthen your understanding. Understanding ICSE Mathematics Class 9 solutions are prepared by Experienced Embibe Experts.