A swarm of locusts may contain as many as 85 million locusts per square kilometer and cover an area of 1200 square kilometers. About how many locusts ...

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Prerequisite Skills

VOCABULARY CHECK

Copy and complete the statement 1. The zeros of the function graphed are ? . ANSWER – 3 and 1

2. The maximum value of the function graphed is ? . ANSWER 4

Prerequisite Skills

VOCABULARY CHECK

3. The standard form of a quadratic equation in one variable is ? where a = 0. ANSWER y = ax2 + bx + c

Prerequisite Skills

SKILLS CHECK

Graph the function. Label the vertex and the axis of symmetry. 4. y = – 2(x – 1)2 + 4 ANSWER 5. y = 3(x – 2)(x + 3) ANSWER

6. y = – x2 – 4x + 4 ANSWER

Prerequisite Skills

SKILLS CHECK

Factor the expression. 7. x2 + 9x + 20

ANSWER (x + 4)(x + 5)

8. 2x2 + 5x – 3

ANSWER (2x + 1)(x – 3)

9. 9x2 – 64

ANSWER (3x – 8)(3x + 8)

Prerequisite Skills

SKILLS CHECK

Solve the equation. 10. 2x2 + x + 6 = 0

ANSWER

– 1 + i √47 4

11. 10x2 + 13x = 3

12. x2 + 6x + 2 = 20

ANSWER – 1.5, 0.2 ANSWER – 3 + 3 √3

Properties of Exponents Ch. 5, Sec. 1

Objectives: 1) Use properties of exponents to evaluate and simplify exponential expressions. 2) Use powers as models.

a

exponent

n

a is used as a factor n times read >> a to the nth power

base Try this: a.

2 •2 4

5

b.

3 3 3

2

Properties of Exponents Let a and b be real numbers and let m and n be integers

a •a = a m

n

m+ n

m•n

(a ) = a m m m (ab) = a b m n

a

−n

1 = n , a≠0 a

a = 1, a ≠ 0 0

m

a m− n =a , a≠0 n a

FG a IJ H bK

m

m

a = m , b≠0 b

Example 1 5 ( − 3 )( − 3 ) a. b.

( −2)

3 3

2

c. (3 xy ) d.

4

4 −3 • 4 3 −2 2

e.

(3 )

f.

1 x • 5 x

g.

62 • 6 65

3

i.

FG 2 IJ H 3K

−2

EXAMPLE 1

Use scientific notation in real life

Locusts A swarm of locusts may contain as many as 85 million locusts per square kilometer and cover an area of 1200 square kilometers. About how many locusts are in such a swarm?

EXAMPLE 3 a.

b.

c.

Simplify expressions

b–4b6b7 r–2 s3

= b–4 + 6 + 7 = b9 –3

Product of powers property

– 2 )–3 ( r = ( s3 )–3

Power of a quotient property

6 r = –9 s

Power of a power property

= r6s9

Negative exponent property

16m4n –5 4n – 5 – (–5) = 8m 2n–5 = 8m4n0= 8m4

Quotient of powers property Zero exponent property

EXAMPLE 4

Standardized Test Practice

EXAMPLE 5

Compare real-life volumes

Astronomy Betelgeuse is one of the stars found in the constellation Orion. Its radius is about 1500 times the radius of the sun. How many times as great as the sun’s volume is Betelgeuse’s volume? Be·tel·geuse [büt'l jz, büt'l jz] noun

Alpha Orionis

red star in Orion: a bright red variable supergiant star that is the second brightest star in the constellation Orion and the twelfth brightest in the night sky Microsoft® Encarta® 2006. © 1993-2005 Microsoft Corporation. All rights reserved.

EXAMPLE 5

Compare real-life volumes

SOLUTION Let r represent the sun’s radius. Then 1500r represents Betelgeuse’s radius. Betelgeuse’s volume = Sun’s volume

=

4 π (1500r)3 3 4 π r3 3

The volume of a sphere is 4 πr3. 3

4 π 15003r3 3 4 π r3 3

Power of a product property

EXAMPLE 5

Compare real-life volumes = 15003r0 = 15003

Quotient of powers property

1

= 3,375,000,000

ANSWER

Zero exponent property Evaluate power.

Betelgeuse’s volume is about 3.4 billion times as great as the sun’s volume.

GUIDED PRACTICE

for Examples 3, 4, and 5

Simplify the expression. Tell which properties of exponents you used. 5.

x–6x5 x3

SOLUTION x–6x5x3

= x–6x5 + 3

Power of a product property

= x2

Simplify exponents.

GUIDED PRACTICE 6.

for Examples 3, 4, and 5

(7y2z5)(y–4z–1)

SOLUTION (7y2z5)(y–4z–1)

= (7y2z5)(y–4z–1)

Power of a product property

= (7y2 – 4)(z5 +(–1)) Simplify = (7y–2)(z4) = 7z4 y2

Negative exponent property

for Examples 3, 4, and 5

GUIDED PRACTICE 7.

s3

2

t–4

SOLUTION s3

t–4

2

=

s (3)2 (t–4 )2

Power of a product property

=

s6 t–8

Evaluate power.

=

s6t8

Negative exponent property

GUIDED PRACTICE 8.

x4y–2 x3y6

for Examples 3, 4, and 5

3

SOLUTION x4y–2 x3y6

3

(x4)3 (y–2)3 = (x3)3(y6)3 =

x12y–6 x9y18

Power of a powers property

Power of a powers property

= x3y–24

Power of a Quotient property

x3 = 24 y

Negative exponent property

Lesson Close: 1. When multiplying powers, what is the importance having the “same base”? 2. Do the properties of exponents hold for decimals as 5.6 and 3.45?

Assignment: Pg. 333-335 >> 2, 16-48 eoe, 50, 52