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10 Algebra 8

A. RATIONAL NUMBERS: THE SET Q

1. Understanding Rational Numbers

Definition The ratio of an integer to a non-zero integer is called a rational nnumber. The set of rational numbers is denoted by Q. Q = {^ a | a , b  Z , b  0} b

Definition The set of positive rational nnumbers is denoted by Q +.

Q +^ = {^ a^ | a 0 and a , b  , b  0} b b

Definition The set of negative rational numbers is denoted by Q–. Q –^ = {^ a^ | a 0 and a , b  , b  0} b b

2. The Set of Positive Rational Numbers

If a rational number represents a point on the number line on the right side of zero, then it is called a positive rational number.

In short, is a positive rational number if a and b are both positive integers or both nega- tive integers. For example, are positive rational numbers, and denoted by 2. 7

(^2) and – 7 –

a b

3. The Set of Negative Rational Numbers

If a rational number represents a point on the number line on the left side of zero, then it is called a negative rational number. In short, is a negative rational number if a is a positive integer and b is a negative integer, or if a is a negative integer and b is a positive integer.

For example, are negative rational numbers. We can write negative rational

numbers in three ways: (^) – 5 –5^5. 4 4 –

–5 (^) and 5 4 –

a b

12 Algebra 8

We can write every rational number as a repeating or terminating decimal. Conversely, we can write any repeating or terminating decimal as a rational number.

For example, = 0.324242424...

0.6 is a terminating decimal, and 0.324––^ is a repeating decimal. There are some decimals which do not repeat or terminate. For example, the decimals 0. ... 3.141592653 ... =  2.71828 ... = e 1.4142135 ... = ñ 2 do not terminate and do not repeat. Therefore, we cannot write these decimals as rational numbers. We say that they are irrational.

(^3) 0.6, and 321 0. 5 990

Definition A number whose decimal form does not repeat or terminate is called an irrational number. The set of irrational numbers is denoted by Q  or I. Definition The union of the set of rational numbers and the set of irrational numbers forms the set of all decimals. This union is called the set of real nnumbers. The set of real numbers is denoted by R. R = Q  Q 

R = R +^  {0}  R – R +^ is the set of positive real numbers R –^ is the set of negative real numbers

For every real number there is a point on the number line. In other words, there is a one-to-one correspondence between the real numbers and the points on the number line.

The real numbers fill up the number line. We can summarize the relationship between the different sets of numbers that we have described in a diagram. As we know, the set of natural numbers is a subset of the set of whole numbers, the set of whole numbers is a subset of the set of integers, the set of integers is a subset of the set of rational numbers, and the set of rational numbers is a subset of the set of real numbers. This relationship is shown by the dia- gram on the left.

          

     

    

N  W  Z  Q  R Q  R

Real NNumbers

Radicals 13

1. Understanding Square Roots

Remember that we can write a  a as a^2. We call a^2 the square of a, and multiplying a number by itself is called squaring the number. The inverse operation of squaring a number is called finding the square root of the number.

After studying this section you will be able to:

1. Understand the concepts of square root and radical number.
2. Use the properties of square roots to simplify expressions.
3. Find the product of square roots.
4. Rationalize the denominator of a fraction containing square roots.

Objectives

Definition If a^2 = b then a is the square rroot of b ( a  0, b  0). We use the symbol ñ to denote the square root of a number. ñ b is read as ‘the square root of b ’. So if a^22 = b then a = ñ b ( a  b , b  0).

Here are the square roots of all the perfect squares from 1 to 100. 12 = 1  ñ1 = 1 6 2 = 36  ò36 = 6 2 2 = 4  ñ4 = 2 7 2 = 49  ò49 = 7 3 2 = 9  ñ9 = 3 8 2 = 64  ò64 = 8 4 2 = 16  ò16 = 4 9 2 = 81  ò81 = 9 5 2 = 25  ò25 = 5 10 2 = 100  ó100 = 10 The equation x^2 = 9 can be stated as the question, ‘What number multiplied itself is 9?’ There are two such numbers, 3 and –3.

Rule If x  R then

In other words, if x is a non-negative real number, then

if x is a negative real number, then x^2^^ –. x

x^2^  x , and

(^2) | | if^ 0.

• if 0.

x x x x x x

^ 

Radicals 15

EXAMPLE 3

a. ñ 2 ñ8 = ó 2 8 = ò16 = 4 b. c. (^) ò 50 ñ2 = ó 50 2 = ó100 = 10 d.

e.

f. (^10)  90  10  90  900  30

Simplify each of the following. a. ñ 2 ñ 8 b. ñ 7 ñ 7 c. ò 50 ñ 2 d. ò 25 ñ 1 e. ó 576 f. ò 10 ò 90

Solution

2. Properties of Square Roots

Property For any real number a and b , where a  0, and b  0, ñ a ñ b = ó a  b.

Property For any real numbers a and b , where a  0, and b > 0, a a. b b

Mathematics is a universal language.

b  (^0) óa. 6 = a  (^0)  ña. ñ 6

Note

If a  0 then ñ a  ñ a = ó a  a = a^^2  a.

For example,

2 2

36 36 6 ( 0), and 5 5 5 5 25 5.

a a a a

For example, (^24 24 4) 2, and 6 6 (^1 ). 49 49 7

If a > 0 then

a a 1 1. a a

= = =

16 Algebra 8

EXAMPLE 4

a. b.

c. d.

e. f.

g.

h.

i. ^      (^3)  3 3  3 2 ^22  2

x y (^) x y 1 1 1 x y x^ y^ x^ y x y xy

2 2 (^ )

a b (^) a b (^) a b a b a b ab ab

a a (^) a a a a a

– 1  – 1 –^1

Simplify the expressions.

a. b. c. d. e. f.

g. h. i.  (^3)  3

x y x y

2

a b ab

24a^3 6a

Solution

Property

For any real number a and n  Z , ( a  0).

Proof

n factors of ñ a n factors of a

( a ) n^  a  a  a  ...  a  a  a  a  ... a  an

( a ) n^  an

For example, (^)       

2 2 3 3 8 8

a a a and

18 Algebra 8

EXAMPLE 6

a.

b.                

2 2 2 2 2

Simplify the expressions. a. ñ8 + 2ò32 – ò18 + ò72 – ò 98 b. 2 ò48 + 3ò27 – ó108 + ó 243

Solution

EXAMPLE 7

a.

b.

c.

d.

e. (^) x y  x y^2

Write the numbers as pure radicals. a. 2 ñ 2 b. 3 ñ 5 c. 5 ñ 3 d. 10ò 10 e. x ñ y

Solution

Property For any non-zero real numbers a , b , c , and x , a ñ x + b ñ x – c ñ x = ( a + b – c )ñ x.

Note

ñ a + ñ b  ó a + b For example, ñ9 + ò16 = 3 + 4 = 7, but ó9 + 16 = ò25 = 5.

Radicals 19

EXAMPLE 8

a. (^) ñ3 + ñ3 = (1 + 1)ñ3 = 2ñ 3 b. 2 ñ5 + ñ5 = (2 + 1)ñ5 = 3ñ 5 c. 3 ñ6 + 4ñ6 = (3 + 4)ñ6 = 7ñ 6 d. 10ñ5 – 3ñ5 = (10 – 3)ñ5 = 7ñ 5

e. ò50 + ò98 + ó162 = ó 25 2 + ó 49 2 + ó 81 2 = 5ñ2 + 7ñ2 + 9ñ 2

= (5 + 7 + 9)ñ2 = 21ñ 2 f. (^5) ñ x – ò 9 x + ó 64 x = 5ñ x – 3ñ x + 8ñ x = (5 – 3 + 8)ñ x = 10ñ x

Perform the operations. a. ñ3 + ñ 3 b. 2 ñ5 + ñ 5 c. 3 ñ6 + 4ñ 6 d. 10ñ5 – 3ñ 5 e. ò50 + ò98 + ó 162 f. 5 ñ x – ò 9 x + ó 64 x

EXAMPLE 9 Compare the following numbers.

a. ñ7 ... 3 b. 3 ñ5 ... 2ò 10 c. 2 ñ7 ... 3ñ 3 d. –2ñ3 ... –3ñ 2

Solution

a. b. c. d.    

2 2

2 2

2 2

Solution

Property Let a , b , m , and n be four real numbers, satisfying a = m + n and b = m  n. Then,

Proof

1. In order to verify these expressions, suppose that t = ò m + ñ n. t^2 = (ò m + ñ n )^2 = (ò m + ñ n )  (ò m + ñ n ) = (ò m  ñ n ) + (ò m  ñ n ) + (ñ n  ò m ) + (ñ n  ñ n ) (by the distributive property) = m + (ò m  ñ n ) + (ñ n  ò m ) + n

= m + n + 2ó m  n (by the commutative property)

a b  t^2 = a + 2ñ b  t =

2. We can prove the second part in the same way. Try it yourself.

a  2 b

m  n  a  2 b m – n  a – 2 b ( m  n )

Radicals 21

4. Write each number as a pure radical. a. 5 ñ 3 b. 3 ñ 5 c. 4 ñ 2 d. 2ñ 5 e. a ñ b
5. Perform the operations.

a. b. 5 ñ2 – ñ 8 c. ò27 – ò 48 d.

6. Compare the numbers. a. 3 ñ5 and 2ò 10 b. c. –2ñ5 and –3ñ 3
7. Write each expression in its simplest form. a. b. c. d. e. f.

g. h. i.

Answers

1. a. 2 b. 16 c. 6x d. 6 e. 4 f. 2 g. h. 2. a. 10 b. a 2 + b – c^3 3. a. 3ñ 2 b. 5ñ 2

c. 4ñ 3 d. 2ñ 5 e. 4ñ 3 f. ñ 2 g. –2ñ 2 h. ò3 x 4. a. ò 75 b. ò 45 c. ò 32 d. ò 20 e.

5. a. 3ò30 b. 3ñ2 c. –ñ3 d. 3ñ3 6. a. 3ñ5 > 2ò10 b. c. –2ñ5 > –3ñ3 7. a. ñ2 – 1 b. 2 + ñ 2 c. (^) ñ5 – ñ 2 d. (^) ñ2 + 1 e. (^) ñ5 – 2 f. 2 – ñ 3 g. 4 h. 6 i. 2

a^2  b

–^1

ab

(^1) and 1 2 3

EXAMPLE 11 Simplify the following.

a. b. c. (^1)  1  9 16

Start from the radical on the ‘inside’ of the expression and move outwards. a. Start with ñ9, on the inside, and work outwards.

b.

c. 1  1  9  1  25  1  5  9 ^3 16 16 4 4 2

Solution

22 Algebra 8

EXAMPLE 12 a. Evaluate b.

c. (^) x  x  x  ... 5. Find x.

2 2 2 2 ... a^ a^ a^ a^ ...^ 7. Find^ a.

a.

b. c.    

    

  

2 2

5

x x x

x x x x

x x

2 2

7

a a a a

a a a a

a a a a

a a

2 2

2

2

Let 2 2 2 2 ....

( 2 2 2 2 ... )

2 2 2 2 2 ...

x

x

x

x

x x x x  x x

 2 x x

  

(simplify)

x 2. Therefore, 2 2 2 2 ... 2.

Solution

(take the square of both sides)

(remove a square root)

4. Multiplying Square Roots

To multiply expressions containing square roots, we used the product property of square

roots: ñ a  ñ b = ó a  b. We can also use the distributive property of multiplication over

addition and subtraction to simplify the products of expressions that contain radicals. For example, 2 ñ 8  3 ñ2 = 2  3  ñ 8  ñ 2 = 6ò 16 = 6  4 = 24

Multiply the rational part by the rational part and the radical part by the radical part.

ñ 2  (ñ3 + 2ñ2) = ñ 2  ñ3 + ñ 2  2  ñ 2 = ñ6 + 2  ñ 2  ñ 2 = ñ6 + 2  2 = ñ6 + 4

24 Algebra 8

EXAMPLE 17 Multiply and simplify.

a. (ñ3 + ñ2)  (ñ5 – 1) b. (ñ5 + ñ3)  (ñ7 + ñ2) c. (2ñ3 + 1)  (ñ5 + 1) d. (3ñ2 – 2)  (ñ5 – ñ3)

a. (ñ3 + ñ2)  (ñ5 – 1)= (ñ 3  ñ5) – (ñ 3  1) + (ñ 2  ñ5) – (ñ 2  1) = ò15 – ñ3 + ò10 – ñ 2 b. (^) (ñ5 + ñ3)  (ñ7 + ñ2) = (ñ 5  ñ7) + (ñ 5  ñ2) + (ñ 3  ñ7) + (ñ 3  ñ2) = ò35 + ò10 + ò21 + ñ 6 c. (2ñ3 + 1)  (ñ5 + 1) = (2ñ 3  ñ5) + (2ñ 3  1) + (1  ñ5) + 1 = 2ò15 + 2ñ3 + ñ5 + 1 d. (3ñ2 – 2)  (ñ5 – ñ3)= (3ñ 2  ñ5) – (3ñ 2  ñ3) – (2ñ5 + 2ñ3) = 3ò10 – 3ñ6 – 2ñ5 + 2ñ 3

Solution

5. Rationalizing Denominators

Look at the numbers They are all fractions, and each fraction has an irrational number as the denominator. In math, it is easier to work with fractions that have a rational number as the denominator.

(^1) , 3 , 10 , and 19. 5 2 12 13

Definition Changing the denominator of a fraction from an irrational number to a rational number is called rationalizing tthe ddenominator of the fraction. Rationalizing the denominator does not change the value of the original fraction. To rationalize the denominator, we multiply the numerator and denominator of the fraction by a suitable factor. For example, if the fraction is in the form we multiply both the numerator and the denominator by ñb.

So, Note that and have the same value: they are

equivalent fractions. Look at some more examples: (^3 3 2 3 2 3 2 6 6) , 2 2 2 2 2 2 2 4 2 (^3 3 3 3 3 3 3) 3, and 3 3 3 3 3 3 3 5 3 5 2 3 5 2 3 10 3 10. 2 2 2 2 2 2 2 2 2 2 4

   ^    

   ^ ^   

ab b

a b

a a b a b ab. b b b b b b

a , b

Radicals 25

a.

b.

c.

d. 3 2 – 2^ (3^ 2 – 2)^ (5 – 2^ 5)^3 2 5 – 3^2 2 2 5 – 2 2 5 – 2^2 5 2 5 (5 2 5) (5 – 2 5) 5 – (2 5) 15 2 – 6 10 – 10 – 4 5 15 2 – 6 10 – 10 – 4 5 25 – 20 5

  ^ ^ ^ ^ 

2 2

 ^ ^ ^  ^ ^ ^   

2 2

  ^  ^ ^ ^ ^ 

 ^ ^  

2 2

  ^  ^ ^  ^ ^ 

 ^    

Solution

Definition An expression with exactly two terms is called a binomial expression. Two binomial expressions whose first terms are equal and last terms are opposite are called conjugates, i.e. a + b and a – b are conjugates.

If a  0 and b  0, then the binomials x ñ a + y ñ b and x ñ a – y ñ b are conjugates. We can use conjugates to rationalize denominators that contain radical expressions. For example, let us rationalize ñ3 – ñ 2 is the conjugate of ñ3 + ñ2.

Therefore, we multiply the numerator and the denominator by ñ3 – ñ2 to rationalize the denominator.

2 2

3 2 3 – 2 ( 3 2 ) ( 3 – 2 ) ( 3) – ( 2 ) 3 – 2^1

( a + b )( a – b ) = a^2 – b^2 (ñ a + ñ b )(ñ a – ñ b ) = a – b where a  0 and b  0.

Remark

EXAMPLE 18 Rationalize the denominators.

a. b. c. d. 3 2 – 2 5  2 5

Radicals 27

EXERCISES 1.

4 4.. Write each number as a mixed radical.

a. (^) ñ 8 b. (^) ò 72 c. (^) ó 243

d. 1000 e. ó 125 f. x y^3

2 2.. Simplify the expressions.

a. ñ 3  ñ 3 b. ñ 5  ñ 5 c. ñ 3  ò 12 d. ñ 3  ò 27 e. ò 2 x  ò 8 x f.

g. (^3) a  5 a h. 3 2 x  4  18 xy^2

6 xy  24 xy

1 1.. Evaluate the square roots.

a. ò 36 b. ó 100 c. – ó 121 d. 16 x^2 e. 25  y^2 f. 121  a^4

3 3.. Simplify the expressions.

a. b. c.

d. e. f.

3 3

xy x y

x y x

x x

5 5.. Perform the operations.

a. 3 ñ3 + 2ñ 3 b. 6 ñ5 + ñ 5

c. –5ñx + 5ñx d. ñ6 – 3ñ 6

e. 3 ò18 + 2ò 72 f. ò80 – ó125 + ò 45

g. ò75 + ó108 – ò48 + ò 27

h.

i.

j. (^) 0.9 – 2.7  1.7  0.

9 x^3^  16 x^3 – 4 x 25 x

6 6.. Write each expression in its simplest form.

a. b.

c. d.

e. f.

g. h.

i. f.

k. (^2)  2 4  12

4  15 – 4 – 15 a  2  8 a – a

28 Algebra 8

111 1.. Perform the operations.

a.

b.

c.

d. (^4 2) –^2 3  2 2 3 – 2 2

112 2.. Perform the operations.

a.

b.

c. 1 1 1 ...^1 2 1 3 2 4 3 100 99

110 0.. Rationalize the denominators.

a. b. c.

d. e. f.

g. h. i.

j. k. l. 10 2 21 7 3

7 7.. Simplify the expressions.

a.

b.

c.

d. 13  6  6  9

8 8.. Find x in each equation.

a. b.

c. 3 x  3 x  3 x  ...  9

2 2 2 2 ...  x^3 3 3 3 ...^  x

9 9.. Find the products.

a. ñ 5  (ñ2 + ñ3) b. ñ 7  (1 + ñ7) c. – ñ 2  (ñ3 – ñ8 + 1) d. ñ 2  (ñ8 + ò32) e. ñ 6  (2ñ3 + 3ñ2) f. (3 + ñ5)  (3 – ñ5) g. (^) (2ñ2 – 3)  (2ñ2 + 3) h. (2ñ3+2)  (2ñ3 – 2) i. (ò12 + ñ8)  (ñ3 – ñ2) j. (–ò12 + 2ñ2)  (ñ2 + ñ3) k.

l.

m. 5  2 3  2  3  16 – 9 3