CorrectionKey=NL-C;CA-C Name Class Date 14 . 1 ...images.pcmac.org/SiSFiles/Schools/GA/ButtsCounty/... · Lesson Performance Task Carbon-14 dating is used to determine the age of

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Explore 1 Understanding Integer Exponents

Recall that powers like 3 2 are evaluated by repeating the base (3) as a factor a number of times equal to the exponent (2) . So 3 2 = 3 ∙ 3 = 9. What about a negative exponent, or an exponent of 0? You cannot write a product with a negative number of factors, but a pattern emerges if you start from a positive exponent and divide repeatedly by the base.

Starting with powers of 3:

3 3 =

3 2 =

3 1 =

Dividing a power of 3 by 3 is equivalent to the exponent by .

Complete the pattern:

3 3 _ ÷3 → 3 2 _ ÷3 → 3 1 _ ÷3 → 3 0 _ ÷3 → 3 -1 _ ÷3 → 3 -2

27 _ ÷3 →9 _ ÷3 →3 _ ÷3 → _ ÷3 → _ ÷3 →

3 -1 = 1 _ 3 , 3 -2 = 1 _ 9 = 1 _ 3

Integer exponents less than 1 can be summarized as follows:

Words Numbers Variables

Any non-zero number raised to the power of 0 is 1; 0 0 is undefined 3 0 = 1 (2.4)

0 = 1

x 0 = 1 for x ≠ 0

Any non-zero number raised to a negative power is equal to 1 divided by the same number raised to the opposite, positive power. 3 -2 = 1 _

3 2 = 1 _

9

x -n = 1 _ x n

for x ≠ 0,

and integer n.

Reflect

1. Discussion Why does there need to be an exception in the second rule for the case of x = 0?

Module 14 637 Lesson 1

14 . 1 Understanding Rational Exponents and Radicals

Essential Question: How are radicals and rational exponents related?

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Explore 2 Exploring Rational Exponents

A radical expression is an expression that contains the radical symbol, √

― .

For n √ ― a , n is called the index and a is called the radicand. n must be an integer greater than 1. a can be any real

number when n is odd, but must be non-negative when n is even. When n = 2, the radical is a square root and the index 2 is usually not shown.

You can write a radical expression as a power. First, note what happens when you raise a power to a power.

( 2 3 ) 2 = (2∙2∙2) 2 = (2∙2∙2) (2∙2∙2) = 2 6 , so ( 2 3 ) 2 = 2 3•2 .

In fact, for all real numbers a and all rational numbers m and n, ( a m ) n = a m ∙ n . This is called

the Power of a Power Property.

A radical expression can be written as an exponential expression: n √ ― a = a k . Find the value for k when n = 2.

Start with the equation. √ ― a = a k

Square both sides. ( √ ― a ) = ( a k )

Definition of square root = ( a k ) 2 Equate exponents. 1 =

Power of a power property a 1 = a Solve for k. k =

Reflect

2. What do you think will be the rule for other values of the radical index n?

Explain 1 Simplifying Numerical Expressions with nth RootsFor any integer n > 1, the nth root of a is a number that, when multiplied by itself n times, is equal to a. x = n

√ ― a ⇒ x n = aThe nth root can be written as a radical with an index of n, or as a power with an exponent of 1 _ n . An exponent in the form of a fraction is a rational exponent. n

― a = a

1 _ n The expressions are interchangeable, and to evaluate the nth root, it is necessary to find the number, x, that satisfies the equation x n = a.

Example 1 Find the root and simplify the expression.

64 1 _ 3

Convert to radical. 64 1 _ 3 = 3

√ ― 64 Rewrite radicand as a power. = 3

― 4 3

Definition of nth root = 4

81 1 _ 4 + 9

1 _ 2

Convert to radicals. 81 1 _ 4 + 9

1 _ 2 =

√ ――

+

√ ――

Rewrite radicands as powers. = 4

√

――

4

+ √ ――

2

Apply definition of nth root. = +

Simplify. =



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Your Turn

3. 8 1 _ 3 4. 16

1 _ 2 + 27 1 _ 3

Explain 2 Simplifying Numerical Expressions with Rational Exponents

Given that for an integer n greater than 1, n √

― b = b 1 _ n , you can use the Power of a Power Property to

define b m _ n for any positive integer m.

b m _ n = b

1 _ n ·m b m _ n = b m· 1 _ n

= ( b 1 _ n )

m

Power of a Power Property = ( b m ) 1 _ n

= ( n √

― b ) m Definition of b 1 __ n = n √ ― b m

The definition of a number raised to the power of m __ n is the nth root of the number raised to the mth power. The power of m and the nth root can be evaluated in either order to obtain the same answer, although it is generally easier to find the nth root first when working without a calculator.

Example 2 Simplify expressions with fractional exponents.

27 2 _ 3

Definition of b m _ n 2 7

2 _ 3 = ( 3 √ ― 27 ) 2

Rewrite radicand as a power. = ( 3 √ ― 3 3 ) 2

Definition of cube root = 3 2

= 9

2 5 3 _ 2

Definition of b m __ n 2 5 3 _ 2 = ( √ ― 25 )

Rewrite radicand as a power. = ( √ ――― )

Definition of root = 5 3

=



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Your Turn

5. 3 2 3 _ 5 6. 4

5 _ 2 - 4 3 _ 2

Elaborate

7. Why can you evaluate an odd root for any radicand, but even roots require non-negative radicands?

8. In evaluating powers with rational exponents with values like 2 __ 3 , why is it usually better to find the root before the power? Would it change the answer to switch the order?

9. Essential Question Check-In How can radicals and rational exponents be used to simplify expressions involving one or the other?


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Evaluate: Homework and PracticeEvaluate the expressions.1. 1 0 -2 2. 5 6 -1

3. 2 -4 4. ( 1 _ 3 ) -2

5. (-2) ° 6. 3 ∙ 6 -2

Find the root(s) and simplify the expression.7. 8 1

1 _ 2 8. 12 5 1 _ 3

9. 4 9 1 _ 2 - 4

1 _ 2 10. 1 6 1 _ 4 + 3 2

1 _ 5

• Online Homework• Hints and Help• Extra Practice



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Simplify the expressions with rational exponents.11. 4 9

3 _ 2 12. 8 5 _ 3

13. 2 7 4 _ 3 + 4

3 _ 2 14. 2 5 3 _ 2 + 16

3 _ 2

Simplify the expressions.15. 2 5 - 1 _ 2 16. 8 - 1 _ 3

17. 1 - 2 _ 3 18. 8 2 _ 3 + 8 - 2 _ 3



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19. 25 1 _ 2 _

27 1 _ 3 20. 7· 10 ⁻3

21. ( 1 _ 4 ) ⁻ 3 _ 2 22. 2·3 6 ⁻ 1 _

2 + 6 ⁻1

23. Geometry The volume of a cube is related to the area of a face by the formula V = A 3 _ 2 .

What is the volume of a cube whose face has an area of 100 c m 2 ?



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24. Biology The approximate number of Calories, C, that

an animal needs each day is given by C = 72 m 3 _ 4 , where

m is the animal’s mass in kilograms. Find the number of Calories that a 16 kilogram dog needs each day.

25. Rocket Science Escape velocity is a measure of how fast an object must be moving to escape the gravitational pull of a planet or moon with no further thrust. The escape velocity for the moon is given approximately by the equation

V = 5600 ∙ ( d _ 1000 ) ⁻ 1 _ 2

, where v is the escape velocity in miles per hour and d is the

distance from the center of the moon (in miles). If a lunar lander thrusts upwards until it reaches a distance of 16,000 miles from the center of the moon, about how fast must it be going to escape the moon’s gravity?

26. Multiple Response Which of the following expressions cannot be evaluated?

a. 4 1 _ 2

b. (-4) ⁻ 1 _ 2

c. 4 ⁻2

d. (-4) ⁻2

e. 0 ⁻ 1 _ 2

f. 0 ⁻2



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yH.O.T. Focus on Higher Order Thinking

27. Explain the Error Yuan is asked to evaluate the expression (-8) 2 _ 3 on his exam,

and writes that it is unsolvable because you cannot evaluate a negative number to an even fractional power. Is he correct, and if so, why? If he is not correct, what is the correct answer?

28. Communicate Mathematical Ideas Show that the nth root of a number, a, can be expressed with an exponent of 1 _ n for any positive integer, n.

29. Explain the Error Gretchen thinks she has figured out how to evaluate the square root of a negative number. Explain why her solution is flawed.

( -1 ) 2 ( -1 ) 1 _ 2 = ( -1 )

2· 1 _ 2

= ( -1 ) 0

= 1 Then she solves for ( -1 )

1 _ 2 which is the same thing as √

_ -1 .

( -1 ) 2 ∙ ( -1 ) 1 _ 2 = 1

( -1 ) 1 _ 2 = 1 _

(-1) 2

= 1 _ 1

= 1

But the square root of -1 cannot be 1, since 1 ∙ 1 = 1, not -1. What mistake did she make?



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Lesson Performance Task

Carbon-14 dating is used to determine the age of archeological artifacts of biological (plant or animal) origin. Items that are dated using carbon-14 include objects made from bone, wood, or plant fibers. This method works by measuring the fraction of carbon-14 remaining in an object. The fraction of the original carbon-14 remaining can be expressed by the function, f = 2 ( ⁻ t _ 5700 ) ,

where t is the length of time since the organism died.

a. Fill in the following table to see what fraction of the original carbon-14 still remains after the passage of time.

t t _ 5700 Fraction of Carbon-14 Remaining

0

5700

11,400

17,100

b. The duration of 5700 years is referred to as the “half-life” of carbon-14 because the amount of carbon-14 drops in half 5700 years after any starting point (not just t = 0 years). Verify this property by comparing the amount of remaining carbon-14 after 11,400 years and 17,100 years.

c. Write the corresponding expression for the remaining fraction of uranium-234, which has a half-life of about 80,000 years.



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CorrectionKey=NL-C;CA-C Name Class Date 14 . 1 ...images.pcmac.org/SiSFiles/Schools/GA/ButtsCounty/... · Lesson Performance Task Carbon-14 dating is used to determine the age of