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