CHECK IF THE RADICAL IS RATIONAL OR IRRATIONAL

Tell whether each expression is rational or irrational.

Problem 1 :

-√64

Solution :

-√64

= -√(8 × 8)

= -8

-8/1 is in the form of a/b

So, -√64 is rational.

Problem 2 :

√1600

Solution :

√1600

1600 is in square root. 1600 is a whole number and also it is a perfect square.

So, we have

√1600 = √(40 × 40) = 40

So, √1600 is rational.

Problem 3 :

±√160

Solution :

±√160

The nearest perfect square of 160 is 169, means it is not a perfect square.

±√160 = 12.6491106…. (irrational)

12.6491106 …..  is a non terminating and non repeating decimal.

So, ±√160 is irrational number.

Problem 4 :

√144

Solution :

√144

144 is in square root. 144 is a whole number and also it is a perfect square.

So, we have

√144 = √(12 × 12) = 12

So, √144 is rational.

Problem 5 :

√125

Solution :

√125

√125 is an irrational, because 125 is not a perfect square.

√125 = 11.1803398….. (irrational)

11.1803398….. is a non terminating and non repeating decimal.

So, √125 is irrational number.

Problem 6 :

-√340

Solution :

-√340

-√340 is an irrational, because -340 is not a perfect square.

-√340 = 18.439088….. (irrational)

18.439088…..  is a non terminating and non repeating decimal.

So, -√340 is irrational number.

Problem 7 :

√1.96

Solution :

√1.96

1.96 is in square root. 1.96 is a decimal number and also it is a perfect square.

So, we have

√1.96 = √(1.4 × 1.4) = 1.4

So, √1.96 is rational.

Problem 8 :

-√0.09

Solution :

-√0.09

-√0.09 it can be expressed as a fraction in the form of -9/100.

-9/100

-9/100 is in the form of a/b

So, -√0.09 is a rational.

Problem 9 :

Which statement is not always true?

1) The sum of two rational numbers is rational.

2) The product of two irrational numbers is rational.

3) The sum of a rational number and an irrational number is irrational.

4) The product of a nonzero rational number and an irrational number is irrational.

Solution :

1) Let a = 5 and b = 2/3

a + b = 5 + 2/3

= (15 + 2)/3

= 17/3

So, it is also a rational.

2) Let a = √3 and b = √5

a x b = √3 x √5

= √15 (irrational)

a x b = √3 x √3

= √(3 x 3)

= 3 (rational)

So, this cannot be always true.

3) Let a = √3 (irrational) and b = 2/7 (rational)

a + b = √3 + 2/7

So, it is irrational number.

4) Let a = 2/3 (rational) and b = √3 (irrational)

a x b = (2/3) √3

= 2√3/3

So, it is irrational.

Problem 10 :

(-2 - √3) (-2 + √3) when simplifies is 

a) Positive and irrational       b) Positive and rational

c) Negative and irrational      d) negative and rational

Solution :

(-2 - √3) (-2 + √3)

Comparing with (a + b)(a - b) = a² - b²

Here a = -2 and b = √3

= (-2)² - (√3)²

= 4 - 3

= 1

Positive and rational.

Problem 11 :

Given that following expressions.

I  -5/8 + 3/5      II. 1/2 + √2      III. √5 √5     IV. 3√49

Which expression(s) result in an irrational number?

1) II, only      2) III, only      3) I, III, IV       4) II, III, IV

Solution :

So II is the rational number, option 1) is correct.

Problem 12 :

A teacher wrote the following set of numbers on the board:

a = √20, b = 2.5 and c = √225

explain why a + b is irrational, but b + c is rational

Solution :

a + b = √20  + 2.5

= √(2 x 2 x 5) + 2.5

= 2 √5 + 2.5

√5 is irrational.

Product of rational and irrational is irrational.

Sum of Irrational and rational is irrational.

b + c = 2.5 + √225

= 2.5 + √(15 x 15)

= 2.5 + 15

= 17.5

It can be converted into fraction.

17.5 = 175/10

After simplifying, we get

= 35/2

So, it is rational.