n egative e xponents, r eciprocals, and t he e xponent l aws relating negative exponents to...
TRANSCRIPT
![Page 1: N EGATIVE E XPONENTS, R ECIPROCALS, AND T HE E XPONENT L AWS Relating Negative Exponents to Reciprocals, and Using the Exponent Laws](https://reader036.vdocuments.mx/reader036/viewer/2022082505/56649dbf5503460f94ab3f7d/html5/thumbnails/1.jpg)
NEGATIVE EXPONENTS, RECIPROCALS, AND THE EXPONENT LAWS
Relating Negative Exponents to Reciprocals, and Using the Exponent Laws
![Page 2: N EGATIVE E XPONENTS, R ECIPROCALS, AND T HE E XPONENT L AWS Relating Negative Exponents to Reciprocals, and Using the Exponent Laws](https://reader036.vdocuments.mx/reader036/viewer/2022082505/56649dbf5503460f94ab3f7d/html5/thumbnails/2.jpg)
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
![Page 3: N EGATIVE E XPONENTS, R ECIPROCALS, AND T HE E XPONENT L AWS Relating Negative Exponents to Reciprocals, and Using the Exponent Laws](https://reader036.vdocuments.mx/reader036/viewer/2022082505/56649dbf5503460f94ab3f7d/html5/thumbnails/3.jpg)
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
![Page 4: N EGATIVE E XPONENTS, R ECIPROCALS, AND T HE E XPONENT L AWS Relating Negative Exponents to Reciprocals, and Using the Exponent Laws](https://reader036.vdocuments.mx/reader036/viewer/2022082505/56649dbf5503460f94ab3f7d/html5/thumbnails/4.jpg)
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:
![Page 5: N EGATIVE E XPONENTS, R ECIPROCALS, AND T HE E XPONENT L AWS Relating Negative Exponents to Reciprocals, and Using the Exponent Laws](https://reader036.vdocuments.mx/reader036/viewer/2022082505/56649dbf5503460f94ab3f7d/html5/thumbnails/5.jpg)
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
![Page 6: N EGATIVE E XPONENTS, R ECIPROCALS, AND T HE E XPONENT L AWS Relating Negative Exponents to Reciprocals, and Using the Exponent Laws](https://reader036.vdocuments.mx/reader036/viewer/2022082505/56649dbf5503460f94ab3f7d/html5/thumbnails/6.jpg)
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
![Page 7: N EGATIVE E XPONENTS, R ECIPROCALS, AND T HE E XPONENT L AWS Relating Negative Exponents to Reciprocals, and Using the Exponent Laws](https://reader036.vdocuments.mx/reader036/viewer/2022082505/56649dbf5503460f94ab3f7d/html5/thumbnails/7.jpg)
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
![Page 8: N EGATIVE E XPONENTS, R ECIPROCALS, AND T HE E XPONENT L AWS Relating Negative Exponents to Reciprocals, and Using the Exponent Laws](https://reader036.vdocuments.mx/reader036/viewer/2022082505/56649dbf5503460f94ab3f7d/html5/thumbnails/8.jpg)
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
![Page 9: N EGATIVE E XPONENTS, R ECIPROCALS, AND T HE E XPONENT L AWS Relating Negative Exponents to Reciprocals, and Using the Exponent Laws](https://reader036.vdocuments.mx/reader036/viewer/2022082505/56649dbf5503460f94ab3f7d/html5/thumbnails/9.jpg)
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
![Page 10: N EGATIVE E XPONENTS, R ECIPROCALS, AND T HE E XPONENT L AWS Relating Negative Exponents to Reciprocals, and Using the Exponent Laws](https://reader036.vdocuments.mx/reader036/viewer/2022082505/56649dbf5503460f94ab3f7d/html5/thumbnails/10.jpg)
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
![Page 11: N EGATIVE E XPONENTS, R ECIPROCALS, AND T HE E XPONENT L AWS Relating Negative Exponents to Reciprocals, and Using the Exponent Laws](https://reader036.vdocuments.mx/reader036/viewer/2022082505/56649dbf5503460f94ab3f7d/html5/thumbnails/11.jpg)
REVIEW
![Page 12: N EGATIVE E XPONENTS, R ECIPROCALS, AND T HE E XPONENT L AWS Relating Negative Exponents to Reciprocals, and Using the Exponent Laws](https://reader036.vdocuments.mx/reader036/viewer/2022082505/56649dbf5503460f94ab3f7d/html5/thumbnails/12.jpg)
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