Exercise 1 Page 542

We are given a set of radicals and asked to find which of these are rational numbers. Let's start by recalling the definition of a rational number.

A number is rational when it can be written as the ratio a/b of two integers, where b≠ 0.

Since 0, 1, 4, and 9 are all perfect squares, we can just pick b= 1 and simplify the radical in order to verify this.

Radical	Rewrite	Simplify	a/b
sqrt(0)	sqrt(0^2)	0	0/ 1
sqrt(1)	sqrt(1^2)	1	1/ 1
sqrt(2)	sqrt(2^2)	2	2/ 1
sqrt(9)	sqrt(3^2)	3	3/ 1

For the rest of the numbers, we will show that they are not rational by contradiction. Let's begin by supposing that sqrt(2) is rational. This means that it can be written as the ratio of two integers a and b with no common factors. sqrt(2)=a/b We can square both sides and rearrange this equation.

sqrt(2)=a/b

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Rewrite

RaiseEqn

LHS^2=RHS^2

(sqrt(2))^2=(a/b)^2

PowSqrt

( sqrt(a) )^2 = a

2=(a/b)^2

PowQuot

(a/b)^m=a^m/b^m

2=a^2/b^2

MultEqn

LHS * b^2=RHS* b^2

2b^2=a^2

RearrangeEqn

Rearrange equation

a^2=2b^2

Let's take a closer look to this equation. a^2= 2b^2 The right side of the equation is 2 times an integer. This means that the left side is divisible by 2. a^2 is divisible by 2 Squared integers have the same prime factors as their base, which means that a is also divisible by 2. a is divisible by 2 Now, since a is divisible by 2, then a^2 is divisible by 4, which means that 2b^2 is also divisible by 4. 2b^2 is divisible by 4 This in turn implies that b^2 is divisible by 2, therefore b is also divisible by 2. b is divisible by 2 However, since we assumed that a and b had no common factors, we have reached a contradiction. Therefore, sqrt(2) is not rational. We can do the same for the rest of the radicals while keeping track of prime factors.

Radical	Suppose	Rewrite	Contradiction
sqrt(2)	sqrt(2)=a/b	a^2=2b^2	a and b are divisible by 2
sqrt(3)	sqrt(3)=a/b	a^2=3b^2	a and b are divisible by 3
sqrt(5)	sqrt(5)=a/b	a^2=5b^2	a and b are divisible by 5
sqrt(6)	sqrt(6)=a/b	a^2=6b^2	a and b are divisible by 2 and 3
sqrt(7)	sqrt(7)=a/b	a^2=7b^2	a and b are divisible by 7
sqrt(8)	sqrt(8)=a/b	a^2=8b^2	a and b are divisible by 2