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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

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

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Simplify each expression. a. –3(–11) –3(–11) = 33 The product of two negative numbers is positive. The product of a positive number and a negative number is negative. –6 ( ) = – 3434 18 4 = –4 1212 Write – as a mixed number. 18 4 b. –6 ( ) 3434

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Use the expression –5.5( ) to calculate the change in temperature for an increase in altitude a of 7200 ft. a 1000 –5.5( ) = –5.5 ( ) 7200 1000 a 1000 Substitute 7200 for a. = –5.5(7.2)Divide within parentheses. = –39.6°FMultiply. The change in temperature is –39.6°F. Real- World Example Temperature. You can use the expression 5.5( ) to calculate the changes in the air temperature in degrees Fahrenheit for an increase in altitude a, measured in feet. A hot air balloon starts on the ground and then rises 7200 feet. Find the change in temperature at the altitude of the balloon. a 1000 7200

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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.

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Use the order of operations to simplify each expression. Do you think the answers to a and b will be the same? a. –0.2 4 = –0.0016 Simplify. = 0.0016 Simplify. b. (–0.2) 4 Write as repeated multiplication. –(0.2 0.2 0.2 0.2) = Write as repeated multiplication. (–0.2)(–0.2)(–0.2)(–0.2) = Exponents and Multiplication using Negative Numbers

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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)b.–54 ÷ (–9) The quotient of a positive number and a negative number is negative. = –14 The quotient of a negative number and a negative number is positive. = 6 Rules for Dividing Real Numbers

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Evaluate for p = and r = –. = –2Simplify. = p ÷ rRewrite the equation. prpr = ÷ Substitute for p and – for r. 3232 3434 ( – ) 3232 3434 = Multiply by –, the reciprocal of –. 3232 4343 ( – ) 4343 3434 prpr 3232 3434 For every real number a, there is a multiplicative inverse such that a ∙ = 1. Example: -5 ∙ = 1 Divide real numbers by using the reciprocal of a number. KEEP the 1st term… CHANGE the sign to multiply… FLIP the 2d term …. 1a1a 1a1a 1 -5 Division using Reciprocal (Multiplicative Inverse)

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