Topic 4Real Numbers

8.1.1 Rational Numbers

• To express a fraction as a decimal, divide the numerator by the denominator.

Example 1 (fraction to decimal)Write ¾ as a decimal. ¾ means 3 ÷ 4. The fraction ¾ can be written as 0.75, since 3 ÷ 4 = 0.75.

Example 2 (decimal to fraction)Write -0.16 as a fraction in simplest form. -0.16 = - 16 /100 (0.16 is 16 hundredths.) = - 4 /25 Simplify. The decimal -0.16 can be written as - 4 / 25

Example 3 (Repeating decimal to fraction)Write 8. 22222… as a mixed number in simplest form. Assign a variable to the value 8.222222…. Let N = 8.222… . Then perform the operations on N to determine its value. N = 8.22222…. 10(N) = 10(8.222…) Multiply each side by 10 because 1 digit repeats. 10N = 82.222… Multiplying by 10 moves the decimal point 1 place to the right. -N = 8.222… Subtract N = 8.222… to eliminate the repeating part. 9N = 74 10N - 1N = 9N 9N/9 = 74 /9 Divide each side by 9. N = 8 2/9 Simplify. The decimal 8.2222…. can be written as 8 2/9

8.1.2 Powers and Exponents

• The product of repeated factors can be expressed as a power.

• A power consists of a base and an exponent.

• The exponent tells how many times the base is used as a factor.

Example 1 Write each expression using exponents. 7 7 7 7 7 7 7 7 = 74 The number 7 is a factor 4 times. So, 7 is the base and 4 is the exponent.

Example 2 y y x y x = y y y x x Commutative Property = (y y y) · (x · x) Associative Property = y3 x2 Definition of exponents

Example 3 Evaluate (-6)4. (-6)4 = (-6) (-6) (-6) (-6) Write the power as a product. = 1,296

Example 4 Evaluate m2 + (n - m)3 if m = -3 and n = 2. m2 + (n - m)3 = (-3) 2 + (2 - (-3)) 3 Replace m with -3 and n with 2. = (-3) 2 + (5) 3

Perform operations inside parentheses.= (-3)(-3) + (5)(5)(5) Write the powers as products. = 9 + 125 Add.= 134

8.1.3 Multiply and Divide Monomials

• The Product of Powers rule states that to multiply powers with the same base, add their exponents.

• The Quotient of Powers rule states that to divide powers with the same base, subtract their exponents.

Simplify. Express using exponents.Example 1

23 x 22

23 x 22 = 23 + 2 The common base is 2.= 25 Add the exponents.Example 22s6(7s7)= (2 x 7)(s6 x s7) Commutative and Associative Properties= 14(s6 + 7) The common base is s.= 14s13 Add the exponents.

Example 3Simplify k8 / k .= k8 -1 The common base is k.= k7 Subtract the exponents.

Example 4Simplify (-2)10 x 56 x 63

(-2)6 x 53 x 62 .

= (-2)10-6 x (5)6-3 x (6)3-2

= (-2)4 x (5)3 x (6)1

= 16 x 125 x 6=12,000

8.1.4 Powers of Monomials

• Power of a Power: To find the power of a power, multiply the exponents.

• Power of a Product: To find the power of a product, find the power of each factor and multiply.

Example 1 Simplify (53)6. = 53 · 6 Power of a power = 518 Simplify.

Example 2Simplify (-3m2n4)3.= (-3)3· m2 · 3· n4 · 3 Power of a product= -27m6n12 Simplify.

8.1.5 Negative Exponents

•Any nonzero number to the zero power is 1.

•Any nonzero number to the negative n power is the multiplicative inverse of the number to the nth power.

Example 17−3 = 1 / 73 Definition of negative exponent

Example 2a−4 = 1 / a4 Definition of negative exponent

Example 3Write 1 / 65 as an expression using a negative exponent. 1 / 65 = 6−5

Definition of negative exponent

Example 4x−3 · x5 = x(−3) + 5 Product of Powers = x2 Add the exponents.

Example 5w−5 / w−7 = w−5 − (−7) Quotient of Powers = w2 Subtract the exponents.

8.1.6 Scientific Notation

•Used to simplify writing of extremely large (or small) numbers.

• A number in scientific notation is written as the product of a factor that is at least one but less than ten and a power of ten.

Example 1

Write 8.65 × 107 in standard form. 8.65 × 107 = 8.65 × 10,000,000 107 = 10 ∙ 10 ∙ 10 ∙ 10 ∙ 10 ∙ 10 ∙ 10 or 10,000,000= 86,500,000 The decimal point moves 7 places to the right.Example 2

Write 9.2 × 10–3 in standard form.

9.2 × 10–3 = 9.2 × 0.001 The decimal point moves 3 places to the left.= 0.0092

Example 3

Write 76,250 in scientific notation.

76,250 = 7.625 × 10,000 The decimal point moves 4 places. (So that there is

one whole number)= 7.625 × 104 because you move 4 places to right to get original number.Example 4

Write 0.00157 in scientific notation.

0.00157 = 1.57 × 0.001 The decimal point moves 3 places.(So that there is

one whole number)= 1.57 × 10–3 because you move 3 places to left to get original number.

8.1.7 Compute with Scientific Notation

• You can use the Product of Powers and Quotient of Powers properties to multiply and divide numbers written in scientific notation.

• In order to add/subtract numbers written in scientific notation, exponents must match.

Example 1Evaluate (3.4 × 105)(2.3 × 103). Express the result in scientific notation.(3.4 × 105)(2.3 × 103) = (3.4 × 2.3)(105 × 103) Commutative and Associative Properties= (7.82)(105 × 103) Multiply 3.4 by 2.3.= 7.82 × 105 + 3 Product of Powers= 7.82 × 108 Add the exponents.

Example 2Evaluate (2.325 × 104)(3.1 × 102) . Express the result in scientific notation.(2.325 × 104)(3.1 × 102) = (2.325 / 3.1 ) (104 / 102 ) Associative Property= (0.75) (104 / 102 ) Divide 2.325 by 3.1.= 0.75 × 104 – 2 Quotient of Powers= 0.75 × 102 Subtract the exponents.= 0.75 × 102 Write 0.75 × 102 in scientific notation.= 7.5 × 10 Since the decimal point moved 1 place to the right, subtract 1 from the exponent.

Example 3Evaluate (5.24 × 105) + (8.65 × 106). Express the result in scientific notation.= (5.24 × 105) + (86.5 × 105) Write 8.65 × 106 as 86.5 × 105.= (5.24 + 86.5) × 105 Distributive Property= 91.74 × 105 Add 5.24 and 86.5.= 9.174 × 106 Write 91.74 × 105 in scientific notation.

8.1.8 Roots

• A square root of a number is one of its two equal factors.

• A radical sign, √is used to indicate a positive square root.

• Every positive number has both a negative and positive square root.

ExamplesFind each square root.1. √1 Find the positive square root of 1; 12 = 1, so √1 =

1.

2. - √16 Find the negative square root of 16; (-4)2 = 16, so - √16 = -4.

3. ± √0.25 Find both square roots of 0.25; 0.52 = 0.25, so ± √0.25 = ±0.5.

4. √-49 There is no real square root because no number times itself is equal to -49.

Example 5 Solve a2 = 4/9. a2 = 4/9 Write the equation. a = ± √ 4/9 Definition of square root a = 2/3 or – 2/3 Check 2/3 · 2/3 = 4/9 and (-2/3) (-2/3) = 4/9 . The equation has two solutions, 2/3 and – 2/3