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Transcript
Section 1-6 Multiply and Divide Real Numbers SPI 12B: Identify the reciprocal of a real number Objectives: • 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 The product of 2 positive numbers or 2 negative numbers is positive. Example 2 ∙ 5 = 10 and (-2)(-5) = 10 Numbers with different signs 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) –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 hot 7200 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 1000 Substitute 7200 for a. = –5.5(7.2) Divide within parentheses. = –39.6°F Multiply. The change in temperature is –39.6°F. Evaluate the Expression Evaluate 5rs for r = –18 and s = –5. 5rs = 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 • 0.2 • 0.2) Write as repeated multiplication. = –0.0016 Simplify. b. (–0.2)4 = (–0.2)(–0.2)(–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 1st term… CHANGE the sign to multiply… FLIP the 2d 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.