Negative Exponents, Reciprocals, and The Exponent Laws
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Transcript of Negative Exponents, Reciprocals, and The Exponent Laws
NEGATIVE EXPONENTS, RECIPROCALS, AND THE EXPONENT LAWS
Relating Negative Exponents to Reciprocals, and Using the Exponent Laws
TODAY’S OBJECTIVES
Students will be able to demonstrate an understanding of powers with integral and rational exponents, including:1. Explain, using patterns, why x-n = 1/xn, x ≠ 02. Apply the exponent laws3. Identify and correct errors in a simplification of an
expression that involves powers
RECIPROCALS
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: am x an = am+n, we can see that this rule also applies to powers 5-2 x 52 = 5-2+2 = 50 = 1
Since the product of these two powers is 1, 5-2 and 52 are reciprocals
So, 5-2 = 1/52, and 1/5-2 = 52
5-2 = 1/25
POWERS WITH NEGATIVE EXPONENTS
When x is any non-zero number and n is a rational number, x-n is the reciprocal of xn
That is, x-n = 1/xn and 1/x-n = xn, x ≠ 0
This is one of the exponent laws:
EXAMPLE 1: EVALUATING POWERS WITH NEGATIVE INTEGER EXPONENTS
Evaluate each power: 3-2
3-2 = 1/32 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
EXAMPLE 2: EVALUATING POWERS WITH NEGATIVE RATIONAL EXPONENTS
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 = 1/82/3 = 1/(3√8)2
1/22
1/4 Your turn: Evaluate (9/16)-3/2
(16/9)3/2 = (√16/9)3 = (4/3)3 = 64/27
EXPONENT LAWS
Product of Powers am x an = am+n
Quotient of Powers am/an = am-n, a ≠ 0
Power of a Power (am)n = amn
Power of a Product (ab)m = ambm
Power of a Quotient (a/b)m = am/bm, b ≠ 0
APPLYING THE EXPONENT LAWS
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
EXAMPLE 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 (1.43)(1.44)/1.4-2
1.43+4/1.4-2 = 1.47/1.4-2 = 1.47-(-2) = 1.49
We will also be simplifying algebraic expressions with integer and rational exponents
EXAMPLE 4 Simplify the expression 4a-2b2/3/2a2b1/3
First use the quotient of powers law 4/2 x a-2/a2 x b2/3/b1/3 = 2 x a(-2)-2 x b2/3-1/3
2a-4b1/3
Then write with a positive exponent 2b1/3/a4
Your turn: Simplify (100a/25a5b-1/2)1/2
(100/25 x a1/a5 x 1/b-1/2)1/2
(4a1-5b1/2)1/2 = (4a-4b1/2)1/2
41/2a(-4)(1/2)b(1/2)(1/2) = 2a-2b1/4
2b1/4/a2
REVIEW
ROOTS AND POWERS HOMEWORK
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