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N EGATIVE E XPONENTS, R ECIPROCALS, AND T HE E XPONENT L AWS Relating Negative Exponents to Reciprocals, and Using the Exponent Laws

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T ODAY ’ S O BJECTIVES Students will be able to demonstrate an understanding of powers with integral and rational exponents, including: 1. Explain, using patterns, why x -n = 1/x n, x ≠ 0 2. Apply the exponent laws 3. Identify and correct errors in a simplification of an expression that involves powers

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R ECIPROCALS Any two numbers that have a product of 1 are called reciprocals 4 x ¼ = 1 2/3 x 3/2 = 1 Using the exponent law: a m x a n = a m+n, we can see that this rule also applies to powers 5 -2 x 5 2 = 5 -2+2 = 5 0 = 1 Since the product of these two powers is 1, 5 -2 and 5 2 are reciprocals So, 5 -2 = 1/5 2, and 1/5 -2 = 5 2 5 -2 = 1/25

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P OWERS WITH N EGATIVE E XPONENTS When x is any non-zero number and n is a rational number, x -n is the reciprocal of x n That is, x -n = 1/x n and 1/x -n = x n, x ≠ 0 This is one of the exponent laws:

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E XAMPLE 1: E VALUATING P OWERS WITH N EGATIVE I NTEGER E XPONENTS Evaluate each power: 3 -2 3 -2 = 1/3 2 1/9 (-3/4) -3 (-3/4) -3 = (-4/3) 3 -64/27 We can apply this law to evaluate powers with negative rational exponents as well Look at this example: 8 -2/3 The negative sign represents the reciprocal, the 2 represents the power, and the 3 represents the root

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E XAMPLE 2: E VALUATING P OWERS WITH N EGATIVE R ATIONAL E XPONENTS Remember from last class that we can write a rational exponent as a product of two or more numbers The exponent -2/3 can be written as (-1)(1/3)(2) Evaluate the power: 8 -2/3 8 -2/3 = 1/8 2/3 = 1/( 3 √8) 2 1/2 2 1/4 Your turn : Evaluate (9/16) -3/2 (16/9) 3/2 = ( √16/9) 3 = (4/3) 3 = 64/27

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E XPONENT L AWS Product of Powers a m x a n = a m+n Quotient of Powers a m /a n = a m-n, a ≠ 0 Power of a Power (a m ) n = a mn Power of a Product (ab) m = a m b m Power of a Quotient (a/b) m = a m /b m, b ≠ 0

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A PPLYING THE E XPONENT L AWS We can use the exponent laws to simplify expressions that contain rational number bases When writing a simplified power, you should always write your final answer with a positive exponent Example 3: Simplifying Numerical Expressions with Rational Number Bases Simplify by writing as a single power: [(-3/2) -4 ] 2 x [(-3/2) 2 ] 3 First, use the power of a power law: For each power, multiply the exponents (-3/2) (-4)(2) x (-3/2) (2)(3) = (-3/2) -8 x (-3/2) 6

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E XAMPLE 3 Next, use the product of powers law (-3/2) -8+6 = (-3/2) -2 Finally, write with a positive exponent (-3/2) -2 = (-2/3) 2 Your turn : Simplify (1.4 3 )(1.4 4 )/1.4 -2 1.4 3+4 /1.4 -2 = 1.4 7 /1.4 -2 = 1.4 7-(-2) = 1.4 9 We will also be simplifying algebraic expressions with integer and rational exponents

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E XAMPLE 4 Simplify the expression 4a -2 b 2/3 /2a 2 b 1/3 First use the quotient of powers law 4/2 x a -2 /a 2 x b 2/3 /b 1/3 = 2 x a (-2)-2 x b 2/3-1/3 2a -4 b 1/3 Then write with a positive exponent 2b 1/3 /a 4 Your turn: Simplify (100a/25a 5 b -1/2 ) 1/2 (100/25 x a 1 /a 5 x 1/b -1/2 ) 1/2 (4a 1-5 b 1/2 ) 1/2 = (4a -4 b 1/2 ) 1/2 4 1/2 a (-4)(1/2) b (1/2)(1/2) = 2a -2 b 1/4 2b 1/4 /a 2

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R EVIEW

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R OOTS AND P OWERS H OMEWORK Page 227-228 #3,5,7,9,11,15,17-21 Extra Practice: Chapter Review, pg. 246 – 249 Review: Chapter 1-4, pg. 252 – 253 Finish Chapter 4 Vocabulary Book

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