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MATH 8 UNIT 2 EXPONENTS
5. Which expression is equivalent to
Test Review
a.
Multiple Choice
1. Which of the following numbers is an irrational
number?
A.
Hint: Irrational numbers
are non-perfect squares,
decimals that never end,
and the number pi.
11
8
B.
3
C
9
xyz
6 y3 z
b.
1
6xy 3 z
c.
1
xy 3 z
d.
x
6 y3 z
4 x 2 yz
?
24 xy 4 z 2
Hint: List out all of the
x,y, and z’s for the top
and the bottom.
Cross out the ones
that they have in
common. Simplify the
whole numbers.
D. –3
2. Which list contains only rational numbers?
(x-4)(x-7)(x5)
A. π, 1.6, 3.4
B. –2, 1,
2
1
C. –4, ,
3
D.
5
1
, 0, 2.66
2
3. Simplify y 2  3y 3
a.
b.
c.
d.
3y 5
3y 6
4 y5
4 y6
6. Which of the following is a simplified form of
this expression?
Hint: Rational numbers
are perfect squares,
decimals that end,
decimals that repeat,
whole numbers, and
fractions.
Hint: List out all of the y’s
and all of the whole
numbers. Then multiply the
whole numbers and add up
the y’s to determine what
the exponent will be.
EX: y4 = y∙y∙y∙y
a. x-5
b. x140
c.
x5
x 11
d.
1
x6
3
 2x  y2
7. Simplify:  
 y  24
a.
b.
x3
3y
c.
x3 y
3
d.
x3
6 y3
6
y
y2
a. 3
b. 4
c. y 3
4. Simplify
d.
y4
Hint: List out all of the y’s
and cancel out what they
have in common.
x3
12 y
Hint: List out all of the x’s
and cross off what they
have in common.
Remember, you can not
keep negative exponents.
You must make it positive
and send it to the
denominator.
Hint: List out all the
whole numbers, x’s and
y’s next to each other
since they are being
multiplied. Then cancel
out what they have in
common. Also, simplify
the whole numbers.
(m 2n 3 )2
p 3
8. Simplify
4
12. Solve the following equation for m.
–3m + 7 = 31
Hint: Multiply the
exponents on top by
2. Then Move any
negative ones to the
opposite (top or
bottom) of where they
are. List out all the
letters, and cancel
what they have in
common.
1
a.
m p
n1
b.
m4 p6
n6
m4 p3
c.
n6
d.
m 4n
p
9. Calculate
a.
b.
c.
d.
3
8
4
16
32
64 .
Hint:
m = –8
m = –3
m = 24
m = 27
1. Move any number not
attached to the variable.
2. Get the variable by itself.
13. Solve the following equation for f.
2(5f – 11) = 28
a.
b.
c.
d.
f=3
f=4
f=5
f=7
Hint: 1. Distribute the 2 into the
parentheses. 2. Move the
number not attached to the
variable to the other side. 3.
Get the variable by itself.
14. . Solve the following equation for x.
Hint: What number times
itself 3 times will give us
64?
4.07  108
4.07  10 8
407  10 6
407  10 6
Hint: Move the decimal to the
right until the number you get is
> 1 and < 10. The exponent will
be the amount of spaces the
decimal was moved. It will be
positive since the standard
form number is
LARGE.
11. Which shows 0.004 written in scientific
notation?
a. 0.4  10 2
b. 4  10 3
c. 4  103
2
d. 0.4  10
1
(18x + 12) = –3x + 40
3
a. x = –21
10. One year it was estimated that 407,000,000
soccer balls were sold worldwide. What is this
number in scientific notation?
a.
b.
c.
d.
a.
b.
c.
d.
Hint: Move the
decimal to the right
until the number you
get is > 1 and < 10. The
exponent will be the
amount of spaces the
decimal was moved.
The exponent will be
negative since the
standard form number
is SMALL.
Hint: Distribute the
b.. x = 4
c. x = 5
d. x = 12
1
first.
3
This basically means to
divide each of the numbers
in the parentheses by 3.
15. Solve for x: 4 x  2( x  3)  10  56
a. x  10
b. x  10
c. x  10.5
2
d. x  8
3
Hint: Distribute the 2 into
the parentheses. Then
combine like terms. Then
get the variable by itself.
Perform the following operations. SHOW
computation (if needed). Write the answer in
scientific notation.
16. ( 3.4  105 ) + ( 7.6 105 )
Hint: Write each
number in
standard form and
then add them
together.
Hint: Multiply the whole
17. ( 5 103 )( 9 10 2 ) numbers. Keep the
multiplication sign and the
10. Add the exponents.
Write it in standard form.
18.
8 102
4 105
24. Why do we need scientific notation? Give
examples from real life and mention the unit of
measure which might be used.
a. a very small number
Hint: Divide the whole
numbers. Keep the
multiplication sign and the
10. Subtract the exponents.
Write it in standard form.
19. (.0721)  (2.68 x102 )
20. (.054)(3.2 x105 )
Hint: Change the 2nd
number to standard
form. Add the two
numbers.
Hint: Change the 2nd
number to standard
form. Multiply the two
numbers together.
21. . The answer on the calculator shows
2.9084E-4. What is the answer in standard
form? Explain what the calculator does.
Hint: The number
after the E tells you
what the exponent is.
2
22. An earthquake of magnitude 3.0 is 10 times
stronger than an earthquake of magnitude 1.0.
An earthquake of magnitude 8.0 is 107 times
stronger than an earthquake of magnitude 1.0.
How many times stronger is an earthquake of
magnitude 8.0 than an earthquake of magnitude
3.0?
Hint: Compare the
exponent amounts.
23. The image of a dust mite from a scanning
electron microscope is 1.5  10 2 millimeters
wide. The image is 5  102 times life size. How
many millimeters wide is the dust mite?
Hint:
Microscope size
Life size
AND Simplify!
b. a very large number
.
25. Is (ab)2 equivalent to ab2? Explain.
Hint: Everything in parentheses
takes the exponent on the outside.
26. Justify why x 0  1 . You may use examples if
needed.
Hint: What does an
exponent of zero mean?
27. Explain the difference between a rational and an
irrational number. Use examples.
28. Examine the following equations. Label each
equation with the following.



No solutions
One solution
Infinite solutions
Hint: No solution, you
are left with a false
statement. (Ex: 3 = 4)
a. 2x  5  x  2  x  3
One solution is when x =
a specific number.
b. 2 x  5  x  3  x  3
Infinite solutions is when
you are left with a true
statement. (Ex: 4 = 4)
c. 3x  5  x  3  x  3
29. Cami and Margaret are saving money. Cami
starts with $7 and saves $13 each week. Maggie
starts with $11 and saves $13 each week. When
will they have the same amount of money? Write
and solve an equation to mathematically prove
your answer.
Hint: Write an expression for each of
them and set them equal to each other
since we want them to be the same.
Answers
1. a
2. d
3. a
4. d
5. d
6. d
7. b
8. c
9. b
10. a
11.b
12. a
13. c
x2
0
Or students may also show that 2  x and
x
x2 x  x
 1 , so x 0  1 .
that 2 
x
xx
27. A rational number can be expressed as a fraction.
Any terminating ( .25 ) or repeating decimal number
( 1.3 ) can be expressed as a fraction (see below—convert
repeating decimal to a fraction).
An irrational number has a non-repeating decimal for
which we can only estimate the fraction (see estimation
process below).
14. b
15. a
6
16. 1.110
1
17. 4.5  10
18. 2 103
19. 9.89 102
20. 1.7280  104
28. a. Infinite. No matter what the value for x, the
21. .00029084 – the calculator’s way of
showing scientific notation
29. 7 + 13x = 11 + 13x; so they will never have the same
amount of money
22. 105
23. 0.3mm
24. A very small number might be
microscopic bacterial growth—probably
measured in millimeters.
A very large number might relate to the
distance from Earth to the sun or to the most
distant star in the universe—probably
measured in miles or kilometers—definitely
involving scientific notation.
25. a 2b 2  ab 2
2
26. x  x  x
x1  x
x0  1
Or students may use numbers.
Or students may use the base 10
pattern.
equation would still be true.
b. No solutions. 5  6 or 0  1
c.One solution. X can have only 1 value in which the
equation will be true.