Download Exponents — Review

Name:
Date:
Algebra 2
Exponents — Review
This homework assignment reviews information about exponents that was covered in prior
courses (Algebra 1 or 1B). Read page 1 for all the facts and properties you will need, do the
problems on pages 2–6, then check your answers using page 7.
Meanings of exponents
Positive whole exponents represent repeated multiplication.
bn = b · b · ··· · b (n b’s)
Powers with exponent 1 are always equal to the base.
b1 = b
Powers with exponent 0 are always equal to 1.
b0 = 1
Negative whole exponents represent fractions.
b–n =
1
bn
If you have a non-whole-number exponent, for now, evaluate on the calculator.
Exponent calculation properties
Multiplying powers with the same base: add the exponents.
am · an = am+n
Dividing powers with the same base: subtract the exponents.
am
= a m -n
n
a
Power of a power: multiply the exponents.
(am)n = amn
Multiplying powers with the same exponent:
an · bn = (ab)n
Dividing powers with the same exponent:
an æ a ö
=ç ÷
bn è b ø
n
Scientific notation
Scientific notation is a way of writing any number as a product of a decimal number and a
power of 10. It’s especially useful for representing numbers that are very large or very small.
Remember that multiplying by powers of 10 can be done easily by moving the decimal point
by the appropriate number of places to the left or to the right. For example:
number in scientific notation
1.35 · 1018
1.35 · 10–7
same number in ordinary notation
1350000000000000000
0.000000135
Your graphing calculator sometimes displays numbers in scientific notation,
especially when the answer to a calculation is very large or very small.
The calculator uses the letter E to say “10 to the power…” See the screen
picture for two examples. Here is what the calculator is actually saying:
15000 · 25000 = 3.75 · 108 = 375000000.
0.0032
= 6.4 · 10–5 = 0.000064.
50
You can also enter numbers into your calculator using scientific notation. When you need to say
“10 to the power…” type (2nd comma), which is labeled EE on your keyboard but displays on
the screen as a single letter E.
Name:
Date:
Problems
1. Using exponent properties, simplify each of these expressions as much as possible.
a. x4 x7
b. (x4)7
c. x · x2 · x3
d. (x2)3 · (x5)4
e. x3 x5 + x4
f. a4 b3 a5 b2
g. a2 b2 (ab)3
h. r4 r – s2 s3
i. (x2 y4 z5) · (x5 y3 z)
Algebra 2
Name:
Date:
Algebra 2
2. Which of the following expressions are equal to x8 ? Circle them.
x4 · x2
x3 · x5
x4 · x4
x2 · x2 · x2
x8 · x
x8 · x0
(x2)3
(x4)2
(x6)2
14 · x4
18 · x8
x8 · y0
(–1)8 · x8
–x8
(–x)8
3. Without a calculator, find these numbers.
a. 0 5
b. 50
c.
(- 43 )2
d.
23
2 -4
4. Convert these numbers from scientific notation into ordinary number notation.
The first one is done for you as an example.
a. 2.468 · 10–8
0.00000002468
b. 2.468 · 108
c. 3.14159 · 10–6
d. 5 · 1013
Name:
Date:
Algebra 2
e. this calculator output: 1.3579 E-6
f. this calculator output: 6.02 E23
(By the way, this is a famous number, it’s the number of atoms in one gram of hydrogen!)
5. Here are some facts about the Earth. Rewrite each of the numbers in scientific notation.
The first one is done for you as an example.
a. Earth’s population:
6,215,000,000 people
6.215 · 109
b. Earth’s volume:
1,083,000,000,000,000,000 square kilometers
c. Earth’s land area:
58,969,045,000,000 square meters
d. If you fill a bucket with dirt, the portion of the whole Earth that’s in the bucket:
0.0000000000000000000000016
6. Do these function evaluations without a calculator. Show your work.
a. Evaluate f(x) = 4 · 5x when x = 3.
b. Evaluate f(x) = 1000 · (1.06)x when x = 0.
c. Evaluate f(x) =
(23 )x
when x = 3.
d. Evaluate f(x) = 80 × (12 )
x
when x = 4.
Name:
Date:
Algebra 2
7. Do these function evaluations with a calculator. If you get a number in scientific notation,
turn it into an ordinary number.
a. Evaluate f(x) = 8.3 · 10x when x = 6.
b. Evaluate f(x) = 500,000 · (1.23)x when x = 17.5.
c. Evaluate f(x) = 2 · (53 )
x
when x = 20.
8. Decide whether each statement is a correct statement about exponents. Circle true or false.
(Assume that all letters stand for positive whole numbers.)
a. c x × c y = c xy
true
false
b. c x × c y = c x+ y
true
false
c. c x + c y = c x + y
true
false
d. c x × c y × c z = c x + y + z
true
false
e. c 0 = 0
true
false
c1 = 0
true
false
f.
9. Rewrite each of the following as a power of x. That is, all of your answers should look like xn
for some number n (which could be positive, zero, or negative).
a. x 4 × x 5
b. x 4 × x 9
c. x8 ·
d. x12 ·
1
x5
1
x8
Name:
Date:
e.
Algebra 2
( x 3 )4
( x 5 )2
æ x 6 ö4
f. çç 2 ÷÷
èx ø
g.
x2
x2
h. x0 · x · x 3
Answers
1. a. x11
f. a9b5
b. x28
c. x6
g. a5b5 or (ab)5
2. These should have been circled:
x3 · x5
4. b. 246800000
h. r5 – s5
i. x7y7z6
x8 · y0
(–x)8
(–1)8 · x8
b. 1
e. x8 + x4
x4 · x4
x8 · x0
(x4)2
18 · x8
3. a. 0
d. x26
c. 9/16
d. 128
c. 0.00000314159
e. 0.0000013579
d. 50000000000000
f. 602000000000000000000000
5. b. 1.083 · 1018
c. 5.8969045 · 1013
d. 1.6 · 10–24
6. a. 500
b. 1000
d. 5
7. a. 8300000
b. 18720155.613
(Answers to b and c were rounded.)
c. 8/27
c. 0.0000731
8. only b and d were true
9. a. x9
f. x16
b. x13
c. x3
g. x0 (=1)
h. x4
d. x4
e. x2