8-sinf. Algebra. Alimov Sh. A, Xolmuhamedov O. R, Mirzaahmedov M. A

Therefore, 8 is a solution. 33

-Algebra Class 8 (Zambak)-Zambak Publishing

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. a is a positive rational number if a and b are both positive integers or both negab tive integers. 2 –2 2 and For example, are positive rational numbers, and denoted by . 7 –7 7

The set of positive rational numbers is denoted by Q+. Q+ =

a a |  0 and a, b  , b  0> b 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. a In short, is a negative rational number if a is a positive integer and b is a negative integer, b or if a is a negative integer and b is a positive integer. –5 5 are negative rational numbers. We can write negative rational and 4 –4 5 –5 5 numbers in three ways: –   . 4 4 –4

The set of negative rational numbers is denoted by Q–. a a Q– = < |  0 and a, b  , b  0>b b 10

A. THE SET OF REAL NUMBERS 1. Understanding Real Numbers In algebra we use many different sets of numbers. For example, we use the natural numbers to express quantities of whole objects that we can count, such as the number of students in a class, or the number of books on a shelf. The set of natural numbers is denoted by N. N = -6

The set of whole numbers is the set of natural numbers together with zero. It is denoted by W. W = -6

The set of integers is the set of natural numbers, together with zero and the negatives of the natural numbers. It is denoted by Z. Z = <. –5, –4, –3, –2, –1, 0, 1, 2, 3, 4, 5, . >-6

We use integers to express temperatures below zero, distances above and below sea level, and increases and decreases in stock prices, etc. For example, we can write ten degrees Celsius below zero as –10°C. To express ratios between numbers, and parts of wholes, we use rational numbers. 8 2 3 0 17 For example, , , – , , and are rational numbers. 3 5 7 7 1 The set of rational numbers is the set of numbers that can be written as the quotient of two integers. It is denoted by Q. Q=

a | a, b   and b  0> b 7 2 -7

Some rational numbers

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. 3 321  0.6, and  0.324 = 0.324242424. 5 990 –– 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

R = R+   R– R+ is the set of positive real numbers R– is the set of negative real numbers

do not terminate and do not repeat. Therefore, we cannot write these decimals as rational numbers. We say that they are irrational.

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 numbers. The set of real numbers is denoted by R. R = Q  Q 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.

-1 0 1 2 Some rational numbers

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 diagram on the left. Algebra 8

1. Understanding Square Roots Remember that we can write a  a as a2. We call a2 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. Objectives

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. Definition

If a2 = b then a is the square root of b (a  0, b  0). We use the symbol ñ to denote the square root of a number. ñb is read as ‘the square root of b’. So if a2 = b then a = ñb (a  b, b  0). Here are the square roots of all the perfect squares from 1 to 100. 

The equation x = 9 can be stated as the question, ‘What number multiplied itself is 9?’ There are two such numbers, 3 and –3. Rule

 x if x  0. x2 | x |   – x if x  0.

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

32  3, ( 32  9  3), and (–3)2  –(–3)  3 ( (–3) 2  9  3).

We can conclude that the square root of any real number will always be greater than or equal to zero. ò–9 is undefined. Negative numbers have no square root because the square of any real number cannot be negative. ò–9  3, since 32 is 9, not (–9). ò–9  –3, since (–3)2 is 9, not (–9).

Note x = ñ9 and x2 = 9 have different meanings in the set of all real numbers.  ñ9 = 32 = |3| = 3  If x2 = 9 then x = 3 or x = –3.

Evaluate each square root. a. ò81

f. ó0.64 = 0.8 i. ò–4 is undefined

–42  –16 is undefined

Evaluate each square root. a. ó100

10000  100 Algebra 8

2. Properties of Square Roots Property

For any real number a and b, where a  0, and b  0, ñañb = óab. For example,

25  16  25  16  5  4  20, 3  27  3  27  81  9, 36 a2  36 a2  6 a ( a  0), and 5  5  5  5  25  5.

Mathematics is a universal language.

Simplify each of the following.

a. ñ2ñ8 = ó28 = ò16 = 4

7  7  7  7  49  7

c. ò50ñ2 = ó502 = ó100 = 10

25  1  25  1  25  5

b. ñ7 ñ7 c. ò50ñ2

576  36  16  36  16  6  4  24

10  90  10  90  900  30

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

If a > 0 then a a

24  4  2, and 6 1 49

Simplify the expressions. a.

25 25 5 = = 9 3 9

16 16 4   49 49 7

625 625 25   144 144 12

1 1 1 – – 100 10 100 24a3

24 a3  4  a2  4  a2  2a 6a

a5  b6  a4  b4  ( a2  b2 )2  a2 b2 ab2

For any real number a and n  Z, ( a )n  an

Proof ( a )n  a  a  a  . 

n factors of ña

For example, ( a )2  a2  a, ( 5)3  53  125, and ( 2 )8  2 8  256  16. 16

Evaluate (ñ2)4 + (ñ5)4 – (ñ5)2 – (ñ2)6. ( 2 )2  ( 5)4 – ( 5)2 – ( 2 ) 6  2 4  5 4 – 5 2 – 2 6  (22 )2  (52 )2 – 52 – (23 )2  22  52 – 5 – 2 3  4  25 – 5 – 8  16

3. Working with Pure and Mixed Radicals Definition

A radical expression is an expression of the form index

Square roots have index 2. However, we usually write square roots in their shorter form, ña: 2

A mixed radical is a radical of the form x  n a (x  Q, x  )

For example, 3ñ2, 6ñ7, and 9ó115 are mixed radicals. ò55, ò99, and ò27 are not mixed radicals. We say that they are pure radicals. We can convert between mixed and pure radical numbers to simplify radical expressions. Property

For any real numbers a and b, where a  0 and b  0, a2  b  a b and a b 

8  4  2  2 2  2  2 2  2  2 2, 27  9  3  32  3  32  3  3 3, 32  16  2  4 2  2  4 2  2  4 2, and 50  25  2  5 2  2  5 2  2  5 2.

Simplify the expressions. a. ñ8 + 2ò32 – ò18 + ò72 – ò98 b. 2ò48 + 3ò27 – ó108 + ó243

8 = 22  2 = 2 2 2 32 = 2 4 2  2 = 8 18 = 3  2 = 3 2 2

72 = 6  2 = 6 2 98 = 7  2 = 7 2 2

8  2 32  18  72  98  2 2  8 2  3 2  6 2 7 2  2  (2  8  3  6  7)  6 2

b. 2 48  3 27 – 108  243  2 4 2  3  3 3 2  3 –

 8 3  9 3 – 6 3 9 3  (8  9 – 6  9) 3  20 3 EXAMPLE

Write the numbers as pure radicals. a. 2ñ2

a. 2 2  2 2  2  2 2  2  b. 3 5  32  5  9  5  c. 5 3  52  3 

d. 10 10  10 2  10  100  10  1000 e. x y  x2 y Property

For any non-zero real numbers a, b, c, and x, añx + bñx – cñx = (a + b – c)ñx .

Note ña + ñb  óa+b For example, ñ9 + ò16 = 3 + 4 = 7, but ó9 + 16 = ò25 = 5. 18

Perform the operations. a. ñ3 + ñ3

e. ò50 + ò98 + ó162 Solution

f. 5ñx – ò9x + ó64x

a. ñ3 + ñ3 = (1 + 1)ñ3 = 2ñ3 b. 2ñ5 + ñ5 = (2 + 1)ñ5 = 3ñ5 c. 3ñ6 + 4ñ6 = (3 + 4)ñ6 = 7ñ6 d. 10ñ5 – 3ñ5 = (10 – 3)ñ5 = 7ñ5 e. ò50 + ò98 + ó162 =

ó252 + ó492 + ó812 = 5ñ2 + 7ñ2 + 9ñ2

= (5 + 7 + 9)ñ2 = 21ñ2 f. 5ñx – ò9x + ó64x = 5ñx – 3ñx + 8ñx = (5 – 3 + 8)ñx = 10ñx EXAMPLE

Compare the following numbers. a. ñ7 . 3

Let a, b, m, and n be four real numbers, satisfying a = m + n and b = m  n. Then, 1.

Proof 1. In order to verify these expressions, suppose that t = òm + ñn. t2 = (ò m + ñn)2 = (ò m + ñn)  (ò m + ñn) = (ò m  ñn) + (òm  ñn) + (ñn  òm ) + (ñn  ñn) (by the distributive property) = m + (ò m  ñn) + (ñn  ò m ) + n = m + n + 2ómn a

(by the commutative property)

 t = a + 2ñb  t = a  2 b 2

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

Simplify the expressions. Use the property to help you. a.

3  2 2  2  1  2 1 2+1

6  32  6  2 8  4  2  2  2 4+2

6 – 4 2  6 – 2  2 2  6 – 2 2 2  2  6 – 2 8  4 – 2 2 – 2 4+2

e. We need a 2 in front of ò21 before we can use the property. Therefore, let us multi2

ply the expression by

Check Yourself 1 1. Simplify the expressions. a. ñ2  ñ2 e.

2 2. Evaluate the following.

a. (ñ3)2 + (ñ4)4 – (ñ5)2 – (ñ2)4

b. (ña)4 + (ñb)2 – (ñc)6

3. Simplify the expressions. a. ò18

f. 2ñ2 + 3ñ2 – 4ñ2 20

g. ò50 – ò18 – ò32

e. 5ñ3 – 2ñ3 + ñ3 h. ó12x + ó27x – ó48x Algebra 8

4. Write each number as a pure radical. a. 5ñ3

5. Perform the operations. a. 6

27 75 12  – 4 4 4

6. Compare the numbers. a. 3ñ5 and 2ò10

7. Write each expression in its simplest form. a.

g. ( 6 – 2 )  ( 8  2 12 )

h. ( 7  1)  ( 8 – 28 )

Answers 1. a. 2 b. 16 c. 6x d. 6 e. 4 f. 2 g. c. 4ñ3 d. 2ñ5 e. 4ñ3 f. ñ2

2. a. 10 b. a2 + b – c3 3. a. 3ñ2 b. 5ñ2

g. –2ñ2 h. ò3 x 4. a. ò75 b. ò45 c. ò32 d. ò20 e. a2  b

1 1 c. –2ñ5 > –3ñ3 7. a. ñ2 – 1  2 3 b. 2 + ñ2 c. ñ5 – ñ2 d. ñ2 + 1 e. ñ5 – 2 f. 2 – ñ3 g. 4 h. 6 i. 2

5. a. 3ò30 b. 3ñ2 c. –ñ3 d. 3ñ3 6. a. 3ñ5 > 2ò10 b.

Simplify the following. a.

Start from the radical on the ‘inside’ of the expression and move outwards. a. Start with ñ9, on the inside, and work outwards. 4  21  13  9  4  21  13  3  4  21  16  4  21  4  4  25  4  5  9  3

1 4 1  6 6 72 16  6 6 72 16 16 4  6 6 72 

 6 6  6  6 36  6  6  36  6

9 25 5 9 3  1  1   16 16 4 4 2 21

a a a a .  7. Find a.

x  x  x  .  5. Find x.

a. Let x  2 2 2 2 . . x2  ( 2 2 2 2 . )2

(take the square of both sides)

(remove a square root)

a a a a .  7 ( a a a a . )2  7 2

( x  x  x  . )2  5 2

x  x  x  x  .  25   5

a a a a .  49    7

4. Multiplying Square Roots To multiply expressions containing square roots, we used the product property of square roots: ña  ñb = óa  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, 2ñ8  3ñ2 = 2  3  ñ8  ñ2 = 6ò16 = 64

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

= 24 ñ2  (ñ3 + 2ñ2) = ñ2  ñ3 + ñ2  2  ñ2 = ñ6 + 2  ñ2  ñ2 = ñ6 + 2  2 = ñ6 + 4 22

Perform the operations. a. ñ2(ñ5 + ñ3)

c. 2ñ5(ñ3 + ñ2 + 2ñ5 – ñ7)

a. ñ2(ñ5 + ñ3) = ñ2  ñ5 + ñ2  ñ3 = ó2  5 + ó2  3 = ò10 + ñ6 b. ñ3(3ñ3 + 2ñ2)= ñ3  3ñ3 + ñ3  2ñ2 = 3  ó3  3 + 2  ó3  2 = 3  3 + 2  ñ6 = 9 + 2ñ6 c. 2ñ5(ñ3 + ñ2 + 2ñ5 – ñ7)= 2ñ5  ñ3 + 2ñ5  ñ2 + 2ñ5  2ñ5 – 2ñ5  ñ7 = 2ò15 + 2ò10 + 4ò25 – 2ò35 = 2ò15 + 2ò10 + 20 – 2ò35

Multiply and simplify. a. (ñ2 + ñ3)  (ñ2 + ñ3)

a. (ñ2 + ñ3)  (ñ2 + ñ3) = ñ2  ñ2 + ñ2  ñ3 + ñ3  ñ2 + ñ3  ñ3 = ñ4 + ñ6 + ñ6 + ñ9 = 2 + 2ñ6 + 3 = 5 + 2ñ6 b. (5 + ñ5)  (5 + ñ5)= 52 + 2  5  ñ5 + (ñ5)2 = 25 + 10ñ5 + 5 = 30 + 10ñ5

Multiply and simplify. a. (ñ2 + 1)  (ñ2 – 1)

b. (ñ5 + ñ3)  (ñ5 – ñ3)

e. (ña + ñb)  (ña – ñb)

c. (1 – 2ñ2)  (1 + 2ñ2)

a. (ñ2 + 1)  (ñ2 – 1) = ñ2  ñ2 – ñ2  1+1  ñ2 – 1  1 = (ñ2)2  12 = 2 – 1 = 1 b. (ñ5 + ñ3)  (ñ5 – ñ3) = (ñ5)2 – (ñ3)2 = 5 – 3 = 2 c. (1 – 2ñ2)  (1 + 2ñ2) = 12 – (2ñ2)2 = 1 – 4  2 = 1 – 8 = –7 d. (ña + 1)  (ña – 1) = (ña)2 – 12 = a – 1 (a  0) e. (ña + ñb)  (ña – ñb) = (ña)2 – (ñb)2 = a – b (a, b  0)

Multiply and simplify. a.

3  5  3 – 5  (3  5)  (3 – 5)  3 2 – ( 5) 2  9 – 5  4  2

2  2  2 – 2  (2  2 )  (2 – 2 )  2 2 – ( 2 ) 2  4 – 2  2

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

5. Rationalizing Denominators 1

Look at the numbers

. They are all fractions, and each fraction 5 2 12 13 has an irrational number as the denominator. In math, it is easier to work with fractions that ,

have a rational number as the denominator. Definition

Changing the denominator of a fraction from an irrational number to a rational number is called rationalizing the denominator 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 a by a suitable factor. For example, if the fraction is in the form , we multiply both the b numerator and the denominator by ñb. a

ab . Note that b

ab have the same value: they are b

equivalent fractions. Look at some more examples: 3 2 3 3

3  10 3 10 .  22 4 Algebra 8

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 xña + yñb and xña – yñb are conjugates. We can use conjugates to rationalize denominators that contain radical expressions. 1 For example, let us rationalize . ñ3 – ñ2 is the conjugate of ñ3 + ñ2. 3 2 Therefore, we multiply the numerator and the denominator by ñ3 – ñ2 to rationalize the denominator. 1 3 2

1  ( 3 – 2) ( 3  2 ) ( 3 – 2 )

3– 2 3– 2   3– 2 3–2 1

(a + b)(a – b) = a2 – b2 (ña + ñb)(ña – ñb) = a – b where a  0 and b  0. EXAMPLE

Rationalize the denominators. a.

5  (3  2 2 ) (3 – 2 2 )(3  2 2 )

5 3  5 2 2 3 2 – (2 2 ) 2

3 5  2 10 3 5  2 10   3 5  2 10 9–8 1

( 3 – 2 ) (2 2  1) 3  2 2  3  1 – 2  2 2 – 2 1   (2 2 – 1) (2 2  1) (2 2 ) 2 – 12

2 6  3–2 2 2 – 2 8 –1

( 6  2 ) (1  3) 6  6  3  2 1  2  3   (1 – 3) (1  3) 12 – ( 3) 2 6  18  2  6 6  3 2  2  6 2 6 4 2   – 6 –2 2 1– 3 –2 –2

(3 2 – 2) (5 – 2 5) 3 2  5 – 3 2  2 5 – 2 5 – 2 2 5   (5  2 5) (5 – 2 5) 5 2 – (2 5) 2

15 2 – 6 10 – 10 – 4 5 15 2 – 6 10 – 10 – 4 5  25 – 20 5 25

Rationalize the denominators to find the sum. 3

2 3–2 2  3 3–2 2   2 3 2 2       3 – 2 2  3 2 2 3–2 2   3–2 2 3 2 2  2

( 3  ( 3 – 2 2 ))  ( 2  ( 3  2 2 )) ( 3  2 2)  ( 3 – 2 2) ( 3  3) – ( 3  2 2 )  ( 2  3) ( 2  2 2 ) ( 3)2 – (2 2 )2 3–2 6  6 4 7– 6 6 –7   3–8 –5 5

Check Yourself 2 1. Rationalize the denominators and simplify. a.

2. Rationalize the denominators and simplify. a.

3. Rationalize the denominators and simplify. a. d.

c. ñ2 d. –ò15 e. ò15 f.

5 –1 3 6 6 d. 2 2

g. 5ñ2 – ò10 + 3 ñ5 – 3 h.

6 –16  3 2 a – ab a  b  2 ab 2 5 j. k. –9ñ3 – 6ñ7 3. a. b. ñ2 c. d. 6 4 3 a–b a–b

e. 17 3 – 3 6 Algebra 8

1. Evaluate the square roots. a. ò36 16×2

5. Perform the operations.

f. ò80 – ó125 + ò45

g. ò75 + ó108 – ò48 + ò27

2. Simplify the expressions. a. ñ3  ñ3

h. 3 2 x  4  18 xy2

9×3  16 x3 – 4 x 25 x

3. Simplify the expressions. 50

72 8 32 x3 y2 24 x

6. Write each expression in its simplest form.

4. Write each number as a mixed radical. a. ñ8

7. Simplify the expressions. a.

10. Rationalize the denominators. a.

10 2 7 –5 10  2 21 7 3

8. Find x in each equation. b. 3 3 3 3 .  x

3x  3x  3x  .  9

9. Find the products. a. ñ5  (ñ2 + ñ3) b. ñ7  (1 + ñ7) c. –ñ2  (ñ3 – ñ8 + 1)

11. Perform the operations. a. b. c. d.

e. ñ6  (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. 28

12. Perform the operations. a.

2 33  2 3–3 5  2 3  2  3  16 – 9 3

1 100  99 Algebra 8

After studying this section you will be able to: 1. Understand the concepts of nth root and rational exponent. 2. Write numbers in radical or rational exponent form. 3. Understand the properties of expressions with rational exponents. 4. Use the properties of rational exponents to solve problems.

A. RATIONAL EXPONENTS 1. nth Roots In section 8.2 we studied exponential equations. For example, 2n  2n = 2 is an exponential equation. Let us solve it. 2n  2n = 22n = 2

22n = 21 2n = 1, n =

1 1 for n in the original equation we will get 2 2  2 2  2, . If we substitute 2 2 1

but we know that ñ2  ñ2 = 2. Therefore, 2 2  2  2 2. Remark

Let x  R – . If xm = xn then m = n. Definition

For any natural number n and a, b  R. 1

n If an = b then a = b n  b. a is called the nth root of b. It is denoted by ñb. n

In the expression ña, a is called the radicand and n is called the index. n

Remember that we do not usually write the index for square roots: ña = ña.

Look at some examples of different roots: 52 = 25 and 5 = ò25

‘the square root of 25 is 5’,

‘the cube root of 8 is 2’,

3 = 27 and 3 = ò27

‘the cube root of 27 is 3’, and

2 = 16 and 2 = ò16

‘the fourth root of 16 is 2’.

2. Rational Exponents Definition

If m and n are positive integers (n > 1), and b is a non-negative real number, n

m is called a rational exponent. n 2

For example, the numbers 8 3 , 4 2 , and 2 2 have rational exponents. EXAMPLE

Write the expressions in radical form.

c. 5 3  3 52  3 25

d. x 4  4 x3 EXAMPLE

Write the expressions with radical exponents. 3

64  43  4 3  41  4

ab2  ( ab 2 )5  a 5  b 5

If b is any real number and n is a positive integer (n > 1): 1. If b > 0 then ñb is a positive real number. n

2. If b = 0 then ñb is zero. n

Check Yourself 3 1. Write the expressions in radical form. 1

2. Write the expressions with rational exponents. 3

3. Simplify the expressions. a. ò16

x2  y4 h. 4 16  a4  b8

Answers 1. a. ñ2 b. 3 a c. 3 a2 d. 5 x3 e. 6 x2  y3 1

f. 6 x g. a2  b2 1

e. x 3 f. a2 g. ( x2  y4 )3 h. ( x3 y2 )6

3. a. 4 b. 3 c. –4 d. 5 e. 2

f. –2 g. xy2 h. 2ab2

B. PROPERTIES OF RADICALS Property

For all real numbers a and b, where a > 0 and b > 0, and for any integer m and n, where m > 1 and n > 1: 1.

Look at the examples of each property. 1.

a. ñ5  ñ4 = ó54 = ò20 5

x3  y6  z2  3 ( x  y2 )3  z2  x  y2 z2

64 x2 4 4 4 4  16 x  4 2 4  x  2 4  x  2 x 4x

a. 2  3  3 2 3  3  3 8  3 

2 4 12 16  27 33

Check Yourself 4 1. Simplify the expressions. a.

2. Perform the operations. a. ñ3  ò12

3. Simplify the expressions. 3

4. Simplify the expressions. a.

5. Perform the operations. a. 32

6. Simplify the expressions. 3

7. Write each expression in its simplest form. 3

Answers 1. a. 2 b. 2 c. –3 d. –2 e. 3x 2. a. 6 b. 3 c. x d. x  3 y2 4. a. 2 3 5 b. 3 3 3 c. 2 4 2 6. a. 12

d. xy 4 xy2 d. 24 x7

5. a. 2 6 2 7. a. 6 3

b. x 12 x c. a 6 a

b. 20 x c. x d. x2 e. 9 81

C. RADICAL EQUATIONS Definition

An equation that has a variable in a radicand is called a radical equation. For example, the equations ñx = 25,

2 x – 1  3 x  5 are radical equations.

Let us look at some examples of how to solve radical equations. EXAMPLE

( x  1)2  32 (take the square of both sides)

Therefore, 8 is a solution. EXAMPLE

( 3x  1)2  52 3x  1  25 3x  24 x8

Therefore, 8 is a solution. 33

Therefore, 11 is a solution.

( x2  12 )2  ( x  6)2

x2  12  x2  12 x  36

Therefore, –2 is a solution. EXAMPLE

Therefore, 41 is a solution. EXAMPLE

4x  1 – 5x – 1  0.

4x  1 – 5x – 1  0 ( 4 x  1)2  ( 5 x – 1) 2 4x  1  5 x – 1 x2

4  2 – 1 – 5  2 – 1 0 ?

Therefore, 2 is a solution. 34

3x  1  3x  6  5.

3x  1  5 – 3 x  6

(take the square of both sides)

( 3x  1)2  (5 – 3 x  6 ) 2

3x  1  25 – 2  5  3 x  6  3 x  6

10  3x  6  30 ( 3 x  6 )2  3 2 3x  6  9 3x  3 x1

3  1  1 – 3  1  6 5 ?

Therefore, 1 is a solution.

Check Yourself 5 1. Solve each equation and check your answer. a. ñx = 5

4x – 3 – 2 x – 2  0

Answer 1. a. 25 b. 4 c. 35 d.  e. 25 f. 8 g. 3 h. Radicals

1 17 i. 2 j. k. 6 l. 3 m. 26 n. 2ñ2 o. 1 p. 1 2 5 4 35

1. Write the expressions in exponential form and simplify if possible. a. ò21 3 5

3. Simplify the expressions. 3

6. Simplify the expressions. 3

x  1 2 x  x 1– 2 x  8 3

4. Perform the operations. a.

2. Write the expressions in radical form. a. a 2

5. Solve the equations.

e. 3x  3–2x  33x+1

g. (x – y)2  (y – x)3  (x – y)–4 h. 298 + 298 + 298 + 298

7. Simplify the expressions. a.

10. Solve the equations for x.

ax –1 ax – 2 5 2 104

e. (2x – 3)3 = (x + 1)3 f. (2x – 4)6 = x6 g. 5  23x – 1 – 23x+1 = 256

8. Perform the operations. a. (–2)5  (–2)3

(–2 3 )–3  (–2 2 )6 1 (– )3 2

12 x  12 x  12 x  81 4x  4x  4x

3  4x – 3  4 x –1 –

3  93 x –1  ((–3)2 ) –4 81  (243) x

11. 3a = 25 and 3b = 5. Find

12. 3x = 4. Find 92x + 3x+1.

13. 2x = 3y = a. Write (12)xy in terms of a, x, and y. 14. 2a – 3 and 4b + 7 are integers with 32a – 3 = 54b + 7. Find a + b.

9. Simplify the expressions. 2

a. 2x – 1 + 2x – 1 + 2x – 1 + 2x – 1

= 1. Find the sum of the possible

b. ax + 2  ax – 3ax c. 3x+1 + 3x – 1 + 3x+2 – 3x – 2

16. 44 = 16x. Find x.

10 x  10 x  10 x  10 x d. 5x  5x  5x  5x

313  311 – 39 312  310

g. (–1)101  (–1)125  (–1)100  (–1)99  (–1)49 h.

((–39)  (–2) )  (–1) (–3)125  6 –127  2126

17. |x + y – 3| + |x – y – 1| = 0. Find x. 18. Simplify |x – 4| + 2|3 – x| if 3 b > c

10. k = 1254  642. How many digits are there in the number k? D) 3x+1

32 x  243x  9x  3. Find x. 81x1

12  48 – 27 75 – 2 3

19. Evaluate C) 3

11  2 30 – 8  2 15 3–2 2

x x2 x  xn . Find n.

After studying this section you will be able to: 1. Define statistics as a branch of mathematics and state the activities it involves. 2. Describe some different methods of collecting data. 3. Present and interpret data by using graphs. 4. Describe and find four measures of central tendency: mean, median, mode, and range.

A. BASIC CONCEPTS 1. What is Statistics? Statistics is the science of collecting, organizing, summarizing and analyzing data, and drawing conclusions from this data. In every field, from the humanities to the physical sciences, research information and the ways in which it is collected and measured can be inaccurate. Statistics is the discipline that evaluates the reliability of numerical information, called data. We use statistics to describe what is happening, and to make projections concerning what will happen in the future. Statistics show the results of our experience. Many different people such as economists, engineers, geographers, biologists, physicists, meteorologists and managers use statistics in their work.

statistics Statistics is a branch of mathematics which deals with the collection, analysis, interpretation, and representation of masses of numerical data. The word statistics comes from the Latin word statisticus, meaning ‘of the state’. The steps of statistical analysis involve collecting information, evaluating it, and drawing conclusions. For example, the information might be about:

what teenagers prefer to eat for breakfast; the population of a city over a certain period; the quality of drinking water in different countries of the world; the number of items produced in a factory. Algebra 8

The study of statistics can be divided into two main areas: descriptive statistics and inferential statistics. Descriptive statistics involves collecting, organizing, summarizing, and presenting data. Inferential statistics involves drawing conclusions or predicting results based on the data collected.

2. Collecting Data We can collect data in many different ways.

a. Questionnaires A questionnaire is a list of questions about a given topic. It is usually printed on a piece of paper so that the answers can be recorded. For example, suppose you want to find out about the television viewing habits of teachers. You could prepare a list of questions such as:

 Do you watch television every day?  Do you watch television: in the morning? in the evening?

 What is your favourite television program?  etc. Some questions will have a yes or no answer. Other questions might ask a person to choose an answer from a list, or to give a free answer. When you are writing a questionnaire, keep the following points in mind:

1. A questionnaire should not be too long. 2. It should contain all the questions needed to cover the subject you are studying. 3. The questions should be easy to understand. 4. Most questions should only require a ‘Yes/No’ answer, a tick in a box or a circle round a choice.

In the example of a study about teachers’ television viewing habits, we only need to ask the questions to teachers. Teachers form the population for our study. A more precise population could be all the teachers in your country, or all the teachers in your school.

Statistics and Graphics

b. Sampling A sample is a group of subjects selected from a population. Suppose the population for our study about television is all the teachers in a particular city. Obviously it will be very Xdifficult to interview every teacher in the city individually. Instead we could choose a smaller group of teachers to interview, for example, five teachers from each school. These teachers will be the sample for our study. We could say that the habits of the teachers in this sample are probably the same as the habits of all the teachers in the city.

A sample is a subset of a population.

The process of choosing a sample from a population is called sampling. The process of choosing a sample from a population is called sampling. When we sample a population, we need to make sure that the sample is an accurate one. For example, if we are choosing five teachers from each school to represent all the teachers in a city, we will need to make sure that the sample includes teachers of different ages in different parts of the city. When we have chosen an accurate sample for our study, we can collect the data we need and apply statistical methods to make statements about the whole population.

c. Surveys One of the most common method of collecting data is the use of surveys. Surveys can be carried out using a variety of methods. Three of the most common methods are the telephone survey, the mailed questionnaire, and the personal interview.

3. Summarizing Data In order to describe a situation, draw conclusions, or make predictions about events, a researcher must organize the data in a meaningful way. One convenient way of organizing the data is by using a frequency distribution table. A frequency distribution table consists of two rows or columns. One row or column shows the data values (x) and the other shows the frequency of each value (f). The frequency of a value is the number of times it occurs in the data set. For example, imagine that 25 students took a math test and received the following marks.

8-sinf. Algebra. Alimov Sh.A., Xolmuhamedov O.R., Mirzaahmedov M.A.

Учебник по алгебре для 8 класса. На узбекском языке. — Toshkent: O’qituvchi, 2010. — 224 b. Umumiy o’rta ta’lim maktablarining 8-sinfi uchun darslik.

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