1. Topic 4 Real Numbers

2. 8.1.1 Rational Numbers To express a fraction as a decimal, divide the numerator by the denominator.

3. 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

4. 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

5. 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.

6. Example 1 Write each expression using exponents. 7 7 7 7 7 7 = 7 4 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 = y 3 x 2 Definition of exponents

7. Example 3 Evaluate (-6) 4. (-6) 4 = (-6) (-6) (-6) (-6) Write the power as a product. = 1,296 Example 4 Evaluate m 2 + (n - m) 3 if m = -3 and n = 2. m 2 + (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. 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.

9. Simplify. Express using exponents. Example 1 2 3 x 2 2 2 3 x 2 2 = 2 3 + 2 The common base is 2. = 2 5 Add the exponents. Example 2 2s 6 (7s 7 ) = (2 x 7)(s 6 x s 7 ) Commutative and Associative Properties = 14(s 6 + 7 ) The common base is s. = 14s 13 Add the exponents.

10. Example 3 Simplify k 8 / k. = k 8 -1 The common base is k. = k 7 Subtract the exponents. Example 4 Simplify (-2) 10 x 5 6 x 6 3 (-2) 6 x 5 3 x 6 2. = (-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

11. 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.

12. Example 1 Simplify (5 3 ) 6. = 5 3 · 6 Power of a power = 5 18 Simplify. Example 2 Simplify (-3m 2 n 4 ) 3. = (-3) 3 · m 2 · 3 · n 4 · 3 Power of a product = -27m 6 n 12 Simplify.

13. 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.

14. Example 1 7 −3 = 1 / 7 3 Definition of negative exponent Example 2 a −4 = 1 / a 4 Definition of negative exponent Example 3 Write 1 / 6 5 as an expression using a negative exponent. 1 / 6 5 = 6 −5 Definition of negative exponent

15. Example 4 x −3 · x 5 = x (−3) + 5 Product of Powers = x 2 Add the exponents. Example 5 w −5 / w −7 = w −5 − (−7) Quotient of Powers = w 2 Subtract the exponents.

16. 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.

17. Example 1 Write 8.65 × 10 7 in standard form. 8.65 × 10 7 = 8.65 × 10,000,000 10 7 = 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

18. 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 × 10 4 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.

19. 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.

20. Example 1 Evaluate (3.4 × 10 5 )(2.3 × 10 3 ). Express the result in scientific notation. (3.4 × 10 5 )(2.3 × 10 3 ) = (3.4 × 2.3)(10 5 × 10 3 ) Commutative and Associative Properties = (7.82)(10 5 × 10 3 ) Multiply 3.4 by 2.3. = 7.82 × 10 5 + 3 Product of Powers = 7.82 × 10 8 Add the exponents. Example 2 Evaluate (2.325 × 10 4 )(3.1 × 10 2 ). Express the result in scientific notation. (2.325 × 10 4 )(3.1 × 10 2 ) = (2.325 / 3.1 ) (10 4 / 10 2 ) Associative Property = (0.75) (10 4 / 10 2 ) Divide 2.325 by 3.1. = 0.75 × 10 4 – 2 Quotient of Powers = 0.75 × 10 2 Subtract the exponents. = 0.75 × 10 2 Write 0.75 × 10 2 in scientific notation. = 7.5 × 10 Since the decimal point moved 1 place to the right, subtract 1 from the exponent.

21. Example 3 Evaluate (5.24 × 10 5 ) + (8.65 × 10 6 ). Express the result in scientific notation. = (5.24 × 10 5 ) + (86.5 × 10 5 ) Write 8.65 × 10 6 as 86.5 × 10 5. = (5.24 + 86.5) × 10 5 Distributive Property = 91.74 × 10 5 Add 5.24 and 86.5. = 9.174 × 10 6 Write 91.74 × 10 5 in scientific notation.

22. 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.

23. Examples Find 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.

24. Example 5 Solve a 2 = 4/9. a 2 = 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