Section 1 6 Multiply and Divide Real Numbers

Section 1 -6 Multiply and Divide Real Numbers Still preparing for RN 3 – Preparing for and utilizing the properties of Real Numbers. Objectives: • I CAN Apply properties of real numbers by multiplying and dividing Identity Property of Multiplication For every real number n, 1 ∙ n = n Multiplication Property of Zero For every real number n, 0 ∙ n = 0 Multiplication Property of (- 1) For every real number n, -1 ∙ n = - n Example 1 ∙ 5 = 5 and 1 ∙ (-5) = -5 Example 35 ∙ 0 = 0 and (-35) ∙ 0 = 0 Example -1 ∙ 5 = -5 and -1 ∙ (-5) = 5

Rules for Multiplying derived from the Properties Numbers with the same sign – Answer is the sign given The product of 2 positive numbers or 2 negative numbers is positive. Example 2 ∙ 5 = 10 and (-2)(-5) = 10 Numbers with different signs – Answer will be negative The product of a positive number and a negative numbers is negative. Example (-2) ∙ 5 = -10 and 6 ∙ (-5) = -30

Simplify each expression. a. – 3(– 11) = 33 The product of two negative numbers is positive. b. – 6 (3) 4 – 6 (3 ) = – 18 The product of a positive number and 4 4 a negative number is negative. = – 4 1 Write – 18 as a mixed number. 4 2

Real- World Example Temperature. You can use the expression a 5. 5(1000 ) to calculate the changes in the air temperature in degrees Fahrenheit for an increase in altitude a, measured in feet. A 7200 hot air balloon starts on the ground and then rises 7200 feet. Find the change in temperature at the altitude of the balloon. Use the expression – 5. 5( a ) to calculate the change in 1000 temperature for an increase in altitude a of 7200 ft. a – 5. 5( ) = – 5. 5 (7200) 1000 Substitute 7200 for a. = – 5. 5(7. 2) Divide within parentheses. = – 39. 6°F Multiply. The change in temperature is – 39. 6°F.

Simplifying Square Roots Positive Radical = Positive Square Root Negative before radical means we’re looking at a negative square root A positive and negative indicates there are 2 roots (1 positive, 1 negative) Fraction, See if you can take the square root of both the numerator and denominator.

Evaluate the Expression Evaluate 5 rs for r = – 18 and s = – 5. 5 rs = 5(– 18)(– 5) Substitute – 18 for r and – 5 for s. = – 90(– 5) 5(– 18) results in a negative number, – 90. = 450 – 90(– 5) results in a positive number, 450.

Exponents and Multiplication using Negative Numbers Use the order of operations to simplify each expression. Do you think the answers to a and b will be the same? a. – 0. 24 = –(0. 2 • 0. 2) Write as repeated multiplication. = – 0. 0016 Simplify. b. (– 0. 2)4 = (– 0. 2)(– 0. 2) = 0. 0016 Write as repeated multiplication. Simplify.

Rules for Dividing Real Numbers Dividing numbers with the same sign The quotient of 2 positive numbers or 2 negative numbers is positive. Example: 6 ÷ 3 = 2 and (-6) ÷ (-3) = 2 Dividing numbers with different signs The quotient of a positive number and a negative numbers is negative. Example: -6 ÷ 3 = -2 and 6 ÷ (-3) = -2 Simplify each expression. a. 70 ÷ (– 5) = – 14 The quotient of a positive number and a negative number is negative. b. – 54 ÷ (– 9) = 6 The quotient of a negative number and a negative number is positive.

Division using Reciprocal (Multiplicative Inverse) For every real number a, there is a multiplicative inverse 1 such that a a ∙ 1 = 1. a Example: -5 ∙ 1 = 1 -5 Divide real numbers by using the reciprocal of a number. KEEP the 1 st term… CHANGE the sign to multiply… FLIP the 2 d term …. Evaluate p for p = 3 and r = – 3. r 2 p =p÷r r 3 3 = 2 ÷ (– 4 3 4 = 2 (– 3 ) = – 2 4 Rewrite the equation. ) 3 3 Substitute 2 for p and – 4 for r. 4 3 Multiply by – 3 , the reciprocal of – 4. Simplify.