Math 10
Name:
UNIT 1 LEARNING GUIDE – EXPONENTS
INSTRUCTIONS:
Using a pencil, complete the following questions as you work through the related lessons.
Show ALL of your work as is explained in the lessons. Do your best and always ask questions
if there is anything that you don’t understand.
1.1 EXPONENTS
1. Rewrite the following using digits.
Ex. Base: 3 Exponent: 5 𝟑𝟑𝟓𝟓
e. Twenty squared
a. Base: 7 Exponent: 4
f. Two cubed
b. Base: 9 Exponent: 3
g. Base: 12 Exponent: 6
c. Four to the power of one
h. Base: 45 Exponent: 2
d. Base: 5 Exponent: 10
i.
Seven to the power of eight
2. Expand.
Ex. 56
𝟓𝟓 × 𝟓𝟓 × 𝟓𝟓 × 𝟓𝟓 × 𝟓𝟓 × 𝟓𝟓
c. 98
d. 251
a. 24
b. 72
3. Evaluate the following using your calculator.
Ex. 46
𝟒𝟒𝟒𝟒𝟒𝟒𝟒𝟒
a. 28
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b. 55
c. 93
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e. 35
d. 254
e. 34
Math 10
4. Insert >, <, or = to make each statement true.
Ex. 26 > 62
c. 203 ____ 104
f. 83 ____ 500
b. 73 ____ 122
e. 101 ____ 52
h. 26 ____ 82
d. 42 ____ 24
a. 33 ____ 27
g. 50 ____ 252
5. Complete the table.
Perfect Squares
Exponent
Expanded
Perfect
Square
2×2
4
12
9
4×4
52
36
7×7
92
64
10 × 10
121
122
6. Circle the perfect squares.
−𝟗𝟗
𝟑𝟑
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𝟑𝟑𝟑𝟑𝟐𝟐
𝟏𝟏𝟏𝟏𝟐𝟐
𝟒𝟒𝟒𝟒𝟒𝟒
𝟏𝟏𝟐𝟐
𝟐𝟐
𝟏𝟏𝟏𝟏
−𝟏𝟏𝟏𝟏𝟏𝟏
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𝟐𝟐. 𝟓𝟓𝟐𝟐
𝟔𝟔𝟔𝟔𝟔𝟔
𝟔𝟔𝟔𝟔
𝟑𝟑𝟑𝟑. 𝟏𝟏
Math 10
1.2 BEDMAS
1. Solve. Reminder: Always use the Order of Operations (BEDMAS) when solving a problem.
18
Ex. 82 − 25
c. 33 + 2
𝟔𝟔𝟔𝟔 − 𝟑𝟑𝟑𝟑
𝟑𝟑𝟑𝟑
d. (17 + 6) − 52
a. 34 − 23
e.
b. 4 × 72
102
22
2. Follow the steps to solve each problem. Reminder: After each operation that you
complete, rewrite what remains in the problem.
Ex. 4 × (12 − 9)3 − 72
Step 1:
Brackets
𝟒𝟒 × 𝟑𝟑𝟑𝟑 − 𝟕𝟕𝟐𝟐
Step 2:
Exponents
Step 3:
Division &
Multiplication
Step 4:
Addition &
Subtraction
a. (31 − 23)2 − 53 × 3
15
b. 3 ×
(44−9)
−5
+ 24
𝟒𝟒 × 𝟐𝟐𝟐𝟐 − 𝟒𝟒𝟒𝟒
𝟏𝟏𝟏𝟏𝟏𝟏 − 𝟒𝟒𝟒𝟒
𝟓𝟓𝟓𝟓
3. Solve.
16
36 2
a. (9 − 2)2 + 8
b.
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81
9
c. (8 + 5 × 5) − � 6 �
× (3 − 2)5
95
d. 4 × 5 − 8 × (9 − 2)2
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Math 10
4. Solve. Reminder: Treat numbers that are in the numerator and denominator of a fraction
as though they are in brackets.
a.
53 −25
b.
18+(4+2)2
c.
2×5
6
× (8 − 3)
42
d. (49 − 57) × 17−8×2
11−2
f. 2 + [9 + (9 − 4)2 ] × 2
52
e. (63 ÷ 9)2 + 16−11 × 4
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100−43
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Math 10
1.3 SQUARE ROOTS OF PERFECT SQUARES
1. Follow the steps to find the square root of each number.
Question:
“What is the
square root
of ___?”
Ex.
Step 1: Find 2 identical Step 2: Use one of
numbers that have a product the numbers from
equal to the number in the Step 1.
square root sign.
Answer
𝟏𝟏𝟏𝟏 × 𝟏𝟏𝟏𝟏
√121
a.
√49
b.
𝟏𝟏𝟏𝟏
How to say the
answer.
“11 is the square
root of 121.”
√100
c.
√4
d.
√144
e.
√81
2. Write out the following using mathematical symbols.
Ex. The square root of 36 is 6.
c. 8 squared is 64.
√𝟑𝟑𝟑𝟑 = 𝟔𝟔
a. The square root of 9 is 3.
d. 12 squared is 144.
b. The square root of 64 is 8.
e. The square root of 196 is 14.
3. Evaluate without a calculator.
Ex. √25
a. √16
b. √64
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5
c. √81
f. √36
e. √144
h. √121
d. √1
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g. √100
Math 10
4. Use prime factorization to find the square root of the perfect squares below.
Ex. √324 = 18
b. √484
a. √256
c. √1225
5. Look at the factors of each number to determine whether or not it is a perfect square. If
the number is a perfect square, circle the factor that is its square root.
a. 128: 1, 2, 4, 8, 16, 32, 64, 128
Perfect Square? Yes No
b. 332: 1, 2, 4, 83, 166, 332
Perfect Square? Yes No
c. 625: 1, 5, 25, 125, 625
Perfect Square? Yes No
d. 171: 1, 3, 9, 19, 57, 171
Perfect Square? Yes No
e. 361: 1, 19, 361
Perfect Square? Yes No
f. 729: 1, 3, 9, 27, 81, 243, 729
Perfect Square? Yes No
6. In your own words, explain why numbers that are perfect squares have an odd number
of factors?
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Math 10
1.4 SQUARE ROOTS OF WHOLE NUMBERS
1. Fill in the missing information on the number line.
2. Determine between which two whole numbers each square root lies. Hint: Use the
number line above to help you.
Ex. √105
Between 10 & 11
a. √57
b. √20
c. √2
d. √34
e. √139
3. Determine between which two square roots of perfect squares each number lies. Hint:
Use the number line above to help you.
Ex. 6.6
b. 7.4
d. 4.9
Between the √𝟑𝟑𝟑𝟑 & √𝟒𝟒𝟒𝟒
a. 2.1
c. 9.3
e. 1.5
4. Use a number line to estimate the value of each square root to the nearest tenth. Use
the “approximately equals” sign (≈) to show that the value is not exact. Reminder: Place
all of the square roots that are between the perfect squares on the number line (with
equal spacing), then use this as a guide to estimate the value of the square root.
Ex. √12
√𝟏𝟏𝟏𝟏 ≈ 𝟑𝟑. 𝟒𝟒
a. √46
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Math 10
b. √21
c. √126
d. √29
5. Use your answers from Question 4 to find the square root of each number to 2 decimal
places. Do not use a calculator. Hint: Find this method in the Examples #1 page of the
lesson Square Roots of Whole Numbers.
Ex. √12
On the number line, the answer looks to be between 3.4 and 3.5.
3.4 × 3.4 = 11.56
3.5 × 3.5 = 12.15 The answer is between 3.4 and 3.5, but closer to 3.5
3.46 × 3.46 = 11.97
3.47 × 3.47 = 12.04 The answer is between 3.46 and 3.47. but is closest to 3.46
√𝟏𝟏𝟏𝟏 = 𝟑𝟑. 𝟒𝟒𝟒𝟒
a. √46
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Math 10
b. √21
c. √126
6. Use your calculator to evaluate. Round your answer to 2 decimal places.
Ex. √88
=
9.38
b. √96
a. √2
d. √75
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c. √500
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e. √20 000
Math 10
1.5 PYTHAGOREAN THEOREM
1. State whether each triangle is a right triangle or not. For each right triangle, identify the
hypotenuse. Reminder: The hypotenuse is the side opposite the right angle.
Ex.
b. d
d. d
This is a right
triangle.
a. d
c. d
e. d
2. Label the sides of the triangles with the letters a, b, and c. Reminder: The hypotenuse
must be labelled “c”.
a.
b.
c.
3. Using numbers 1 – 5, order the steps used to determine the length of a side of a right
triangle using Pythagoras’ Theorem.
____________
____________
____________
Step 1
____________
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Write out the formula: 𝑎𝑎2 + 𝑏𝑏 2 = 𝑐𝑐 2
Add the unit of measure to your answer
Replace the variables 𝑎𝑎, 𝑏𝑏, and/or 𝑐𝑐 with known values
Label the sides of the triangle 𝑎𝑎, 𝑏𝑏, and 𝑐𝑐
Calculate
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Math 10
4. Follow the steps to calculate the length of the hypotenuse.
Ex.
c. a
Step 1: Label sides.
Step 2: 𝒂𝒂𝟐𝟐 + 𝒃𝒃𝟐𝟐 = 𝒄𝒄𝟐𝟐
Step 3: 𝟏𝟏𝟏𝟏𝟐𝟐 + 𝟗𝟗𝟐𝟐 = 𝒄𝒄𝟐𝟐
Step 4: 𝟏𝟏𝟏𝟏𝟏𝟏 + 𝟖𝟖𝟖𝟖 = 𝒄𝒄𝟐𝟐
𝟐𝟐𝟐𝟐𝟐𝟐 = 𝒄𝒄𝟐𝟐
√𝟐𝟐𝟐𝟐𝟐𝟐 = 𝒄𝒄
𝟏𝟏𝟏𝟏 = 𝒄𝒄
Step 5: 𝒄𝒄 = 𝟏𝟏𝟏𝟏 𝒄𝒄𝒄𝒄
a. a
d. a
b. a
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e. a
Math 10
5. Calculate the length of the hypotenuse. Round your answer to the nearest hundredth.
Reminder: Always write out the formula at the start of your calculations. Don’t forget to
add the unit of measure to your answer.
a. s
b. s
c. s
6. Calculate the length of the unknown side. Round your answers to the nearest tenth.
a.
b.
Step 1: Label sides.
Step 2: 𝒂𝒂𝟐𝟐 + 𝒃𝒃𝟐𝟐 = 𝒄𝒄𝟐𝟐
Step 3: 𝟓𝟓𝟐𝟐 + 𝒃𝒃𝟐𝟐 = 𝟕𝟕𝟐𝟐
Step 4: 𝟐𝟐𝟐𝟐 + 𝒃𝒃𝟐𝟐 = 𝟒𝟒𝟒𝟒
−𝟐𝟐𝟐𝟐
− 𝟐𝟐𝟐𝟐
𝟐𝟐
𝒃𝒃 = 𝟐𝟐𝟐𝟐
Step 5:
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𝒃𝒃 = √𝟐𝟐𝟐𝟐
𝒃𝒃 = 𝟒𝟒. 𝟗𝟗𝟗𝟗
𝒃𝒃 = 𝟒𝟒. 𝟗𝟗 𝒎𝒎
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Math 10
7. Which triangle in Question #6 is a Pythagorean triple? Why?
8. Use the Pythagorean Theorem to determine whether each triangle is a right triangle.
Reminder: After substituting in the values for 𝑎𝑎, 𝑏𝑏, & 𝑐𝑐, if 𝑎𝑎2 + 𝑏𝑏 2 = 𝑐𝑐 2 , then you know
that a triangle is a right triangle.
a. s
b.
c.
9. Use the Pythagorean Theorem to solve the problems below. Round your answers to the
nearest tenth. Hint Draw a diagram of the problem as a first step to solving.
a. What is the height of computer screen that has a diagonal measurement of 22
inches and a width of 12 inches?
b. A door jamb has a height of 2.08 m, a width of 1.05 m and a diagonal
measurement of 2.33 m. Are the corners of the door jamb square (ie. 90°)?
c. After take-off from the airport, a plane travels for in a straight trajectory until it
reaches its cruising altitude of 11 000 metres (11 km). At this point, it is directly
over a town that is 200km from the airport. Calculate the actual distance
traveled by the plane.
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Math 10
1.6 CUBES & CUBE ROOTS
1. Circle the perfect cubes.
𝟑𝟑𝟑𝟑𝟑𝟑
𝟐𝟐𝟐𝟐
𝟖𝟖
𝟏𝟏𝟑𝟑
𝟐𝟐
𝟒𝟒. 𝟏𝟏𝟑𝟑
𝟑𝟑
2. Complete the table.
Exponent
13
53
𝟐𝟐𝟐𝟐𝟐𝟐
𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏
−𝟗𝟗𝟓𝟓
𝟏𝟏𝟏𝟏𝟏𝟏
𝟔𝟔𝟔𝟔
𝟑𝟑𝟑𝟑
Perfect Cubes
Expanded
Perfect
Cube
2×2×2
8
4×4×4
10 × 10 × 10
27
216
3. Follow the steps to find the cube root of each number.
Question:
“What is the
cube root of
___?”
Ex.
a.
b.
c.
d.
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3
√125
3
√64
Step 1: Find 3 identical Step 2: Use one of
numbers that have a product the numbers from
equal to the number in the Step 1.
cube root sign.
Answer
𝟓𝟓 × 𝟓𝟓 × 𝟓𝟓
3
√729
3
√343
3
√512
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𝟓𝟓
How to say the
answer.
“5 is the cube root
of 125.”
Math 10
4. Fill in the number line below.
5. Estimate the cube root of the following numbers. Write your answers using 1 decimal
point. Hint: Use the number line above to help with your estimations.
3
Ex. √24
≈
c. √6
3
g. √280
3
h. √530
𝟐𝟐. 𝟗𝟗
d. √120
3
e. √946
3
f.
3
3
3
a. √45
b. √143
3
√2
6. Evaluate using your calculator. Round your answers to the nearest hundredth.
3
Ex. √50
3
a. √400
3
b. √16
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3.68
3
f.
3
g. √36
c. √100
d. √1100
3
e. √2
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3
√15 625
3
3
h. √125 000
Math 10
1.7 EXPONENT LAWS
1. Expand, then simplify. Leave your answers in exponential form. Reminder: A number
without an exponent is the same as that number having an exponent of 1.
b. 26 × 23
Ex. 42 × 45
(𝟒𝟒 × 𝟒𝟒) × (𝟒𝟒 × 𝟒𝟒 × 𝟒𝟒 × 𝟒𝟒 × 𝟒𝟒)
𝟒𝟒 × 𝟒𝟒 × 𝟒𝟒 × 𝟒𝟒 × 𝟒𝟒 × 𝟒𝟒 × 𝟒𝟒
𝟒𝟒𝟕𝟕
c. 54 × 5
a. 74 × 72
2. Simplify. Leave your answers in exponential form. Reminder: To multiply terms with
exponents, you can add the exponents if the terms have the same base.
Ex. 64 × 63
d. 152 × 156
b. 35 × 35 × 32
𝟔𝟔𝟒𝟒+𝟑𝟑 = 𝟔𝟔𝟕𝟕
1 4
3. Expand, then simplify. Leave your answers in exponential form.
66
Ex. 45 ÷ 43
b.
62
c. 146 ÷ 146
a. 107 ÷ 106
4. Simplify. Leave your answers in exponential form. Reminder: To divide terms with
exponents, you can subtract the exponents if the terms have the same base.
69
Ex. 28 ÷ 22
d. 73 ÷ 7
b. 65
𝟐𝟐𝟖𝟖−𝟐𝟐 = 𝟐𝟐𝟔𝟔
10
a. 25
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7
÷ 25
c. 32 ÷ 36
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1 3
e. �3� × �3�
c. 93 × 9
a. 87 × 82
e.
416
415
Math 10
5. Match each expression with its simplified form.
a. 𝟕𝟕𝟏𝟏𝟏𝟏 ÷ 𝟕𝟕𝟏𝟏𝟏𝟏
𝟕𝟕𝟏𝟏𝟏𝟏 × 𝟓𝟓𝟓𝟓
b. 𝟕𝟕𝟐𝟐 × 𝟕𝟕𝟓𝟓 × 𝟕𝟕𝟐𝟐
𝟕𝟕𝟑𝟑
d. 𝟕𝟕𝟕𝟕 ÷ 𝟕𝟕𝟎𝟎
𝟕𝟕𝟖𝟖 × 𝟓𝟓𝟕𝟕
c. 𝟕𝟕𝟒𝟒 × 𝟕𝟕𝟔𝟔 × 𝟓𝟓𝟐𝟐 × 𝟓𝟓𝟑𝟑
(−𝟕𝟕)𝟒𝟒
e. (−𝟕𝟕)𝟗𝟗 ÷ (−𝟕𝟕)𝟓𝟓
𝟕𝟕𝟗𝟗
f. 𝟕𝟕𝟒𝟒 × 𝟕𝟕𝟔𝟔 × 𝟕𝟕𝟐𝟐 × 𝟓𝟓𝟑𝟑
𝟕𝟕𝟓𝟓
g. 𝟕𝟕 × 𝟕𝟕𝟑𝟑 × 𝟕𝟕
𝟕𝟕𝟏𝟏𝟏𝟏 × 𝟓𝟓𝟑𝟑
h. 𝟓𝟓𝟒𝟒 × 𝟕𝟕𝟔𝟔 × 𝟕𝟕𝟐𝟐 × 𝟓𝟓𝟑𝟑
6. Simplify.
a. 𝑥𝑥10 × 𝑥𝑥 3
b. 𝑛𝑛5 × 𝑛𝑛6
𝟕𝟕𝟕𝟕
c. 𝑎𝑎 × 𝑎𝑎 × 𝑎𝑎 × 𝑎𝑎
e.
𝑚𝑚7
d. 𝑦𝑦12 ÷ 𝑦𝑦 8
f.
𝑓𝑓 6
e.
𝑟𝑟 5 𝑠𝑠7 𝑡𝑡 3
f.
𝑎𝑎8 𝑏𝑏 4 𝑐𝑐 2
7. Simplify the expressions as much as possible.
c. (𝑥𝑥 5 𝑦𝑦 3 )(𝑥𝑥 2 𝑦𝑦 4 )
a. 𝑥𝑥 2 𝑦𝑦 3 𝑥𝑥 5 𝑦𝑦 7
b. 𝑎𝑎4 𝑏𝑏 3 𝑎𝑎5 𝑏𝑏 2
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𝑦𝑦 9 𝑧𝑧 4
d. 𝑦𝑦 6 𝑧𝑧 3
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𝑚𝑚2
𝑓𝑓
𝑟𝑟 2 𝑠𝑠3 𝑡𝑡
𝑎𝑎4 𝑏𝑏 3
Math 10
8. Simplify the expressions. Reminder: We can evaluate a number with an exponent, but
not a variable with an exponent, unless we know the value of that exponent.
a.
33 𝑥𝑥 5 𝑦𝑦 2
d.
b.
𝑑𝑑4 𝑥𝑥 5 𝑓𝑓𝑑𝑑7
e.
𝑥𝑥 5 𝑦𝑦 2 𝑥𝑥 4 𝑧𝑧 8
ℎ2 𝑤𝑤 8
f.
8𝑚𝑚10 𝑛𝑛9
c.
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𝑥𝑥 2
𝑑𝑑6 𝑥𝑥 2
52 𝑤𝑤 3 ℎ𝑤𝑤𝑤𝑤
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25 𝑎𝑎6 𝑏𝑏 3
𝑎𝑎2 𝑏𝑏2 𝑎𝑎2
𝑥𝑥 2 𝑧𝑧 3 𝑧𝑧 2
4𝑚𝑚6 𝑛𝑛
Math 10
1.8 EXPONENT LAWS 2
1. Simplify. Reminder: The exponent outside of the brackets is applied to everything in the
brackets.
d. (2𝑛𝑛3 )6
Ex. (2𝑦𝑦 5 )3
𝟐𝟐𝟑𝟑 = 𝟖𝟖
(𝒚𝒚𝟓𝟓 )𝟑𝟑 = 𝒚𝒚𝟓𝟓×𝟑𝟑 = 𝒚𝒚𝟏𝟏𝟏𝟏
𝟖𝟖𝒚𝒚𝟏𝟏𝟏𝟏
a. (𝑥𝑥 4 )5
e. (5𝑥𝑥 7 )2
b. (𝑎𝑎2 𝑏𝑏 3 )5
f. (3𝑎𝑎2 𝑏𝑏)3
c. (𝑑𝑑 6 𝑒𝑒 2 𝑓𝑓 4 )4
g. (4𝑓𝑓 2 𝑔𝑔ℎ10 )4
2. Circle the expressions that are equal to 1. Reminder: Any number or variable to the power
of 0 is equal to 1.
a. 𝟕𝟕𝟎𝟎
𝟎𝟎𝟒𝟒
f.
b. (𝟐𝟐𝒚𝒚)𝟎𝟎
g. 𝟓𝟓𝟓𝟓𝟎𝟎 𝟑𝟑𝟑𝟑𝟐𝟐
c.
𝒙𝒙𝟒𝟒 𝒙𝒙𝟎𝟎
h. 𝒉𝒉𝟎𝟎
d. 𝟐𝟐𝟐𝟐𝟎𝟎
i.
(𝟗𝟗𝟗𝟗𝟓𝟓 )𝟎𝟎
e. 𝟒𝟒𝟒𝟒𝟒𝟒𝟎𝟎
j.
(𝟗𝟗𝟗𝟗𝟎𝟎 )𝟓𝟓
3. Expand, then evaluate. Reminder: Don’t forget to use the rules for multiplying negative
numbers: (−)(−) = (+), (−)(+) = (−)
Ex. −𝟐𝟐𝟑𝟑
c. (−𝟐𝟐)𝟒𝟒
a. (−𝟐𝟐)𝟑𝟑
d. (−𝟐𝟐𝟐𝟐)𝟓𝟓
−(𝟐𝟐 × 𝟐𝟐 × 𝟐𝟐) = −𝟖𝟖
e. (−𝟐𝟐𝟐𝟐)𝟔𝟔
b. −𝟐𝟐𝟒𝟒
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Math 10
4. Determine whether the product will be positive or negative and circle your answer. You do
not have to evaluate the expressions. Reminder: When the negative sign is not inside
brackets, the exponent does not affect the negative.
a. (−𝟐𝟐)𝟑𝟑
POS NEG
c. −𝟐𝟐𝟐𝟐
POS NEG
b. (−𝟐𝟐)𝟒𝟒
POS NEG
d. −(𝟐𝟐𝟐𝟐)𝟐𝟐
POS NEG
e. −𝟐𝟐𝟐𝟐𝟎𝟎
POS NEG
g. (−𝟐𝟐𝟐𝟐𝟓𝟓 )𝟐𝟐
POS NEG
f. −(𝟐𝟐𝟐𝟐)𝟕𝟕
h. (−𝟐𝟐𝟐𝟐𝟓𝟓 )𝟑𝟑
5. Simplify and evaluate.
a. (𝑑𝑑 6 𝑒𝑒 2 )2 (𝑑𝑑𝑒𝑒 3 )5
POS NEG
POS NEG
c. (5𝑚𝑚4 𝑛𝑛2 )3 (𝑚𝑚2 𝑛𝑛0 )4
d. (2𝑎𝑎2 𝑏𝑏 2 )4 (6𝑎𝑎6 𝑏𝑏 2 )0 (−3𝑎𝑎3 )3
b. (𝑥𝑥 2 𝑦𝑦 2 𝑧𝑧 2 )5 (𝑥𝑥 6 𝑦𝑦)3
6. Simplify in order to remove the negative exponents. Leave you answers in exponent form.
Reminder: A negative exponent can be removed by taking the reciprocal (ie. opposite) of
that number.
𝟏𝟏
d. 5𝑥𝑥 −4
b. 𝑦𝑦 −3
Ex. 6−2
a. 4−6
𝟔𝟔𝟐𝟐
c. (5𝑥𝑥)−4
e. 3−8 𝑦𝑦 2
7. Simplify in order to remove the negative exponents. Leave you answers in exponent form.
a.
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1
3−2
b.
1
𝑥𝑥 −2
Page 20 of 31
c.
5
𝑦𝑦 −6
Math 10
d.
3−4
2−2
e.
𝑥𝑥 −2
f.
𝑦𝑦 −8
33
3−2
8. Simplify. Answer in exponent form and with no negative exponents. Reminder: The rules of
multiplying negative and positive numbers applies to exponents as well.
e. (99−9 )9
a. (42 )−3
c. (𝑥𝑥 −5 )0
b. (4−2 )−3
f. (2𝑦𝑦 −2 )3
d. (170 )−6
9. Expand. Do not simplify.
Ex. (3𝑟𝑟 + 1)2
(𝟑𝟑𝟑𝟑 + 𝟏𝟏)(𝟑𝟑𝟑𝟑 + 𝟏𝟏)
c. (3𝑎𝑎 + 2𝑏𝑏 − 6)3
b. (𝑥𝑥 − 3)4
e. (−2𝑥𝑥 2 + 5𝑥𝑥 + 4)2
a. (−4𝑡𝑡 + 5)2
d. (6𝑚𝑚2 𝑛𝑛)4
10. Use all of the skills you have learned so far to simplify the following expressions as much as
possible. Answer without negative exponents.
(9𝑦𝑦)−2
a. (𝑎𝑎6 )3 (𝑎𝑎3 )−4
b.
(40𝑦𝑦 5 )0
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Math 10
1.9 MIXED EXPONENTS
1. Identify the base. Reminder: The base is everything that the exponent is applied to.
Ex. (−4𝑡𝑡 + 5)2 (−𝟒𝟒𝟒𝟒 + 𝟓𝟓)
c. (6𝑎𝑎𝑎𝑎)4
b. (−6)(3𝑣𝑣)−3
e. (63)(−1)(8)10
d. 100𝑥𝑥 3
a. −4𝑡𝑡 + 52
2. Identify the coefficient in each of the following
a. -4t ____________
d. 100___________
b. 14v3 ___________
e.
c.
345a4b4___________
𝑥𝑥 4
10
___________
3. Simplify.
Ex.
a.
2 3
� �
𝑥𝑥
4 2
� �
𝑦𝑦
b.
�
1 6
�
𝑔𝑔ℎ
d.
c.
e.
4. Simplify so that only positive exponents remain in your answer. Hint: Take your time and
keep your work neat.
a.
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5 −3
� �
𝑤𝑤
b.
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Math 10
i. (𝑥𝑥 3+𝑥𝑥 )(2𝑥𝑥 4 )
c.
3𝑥𝑥−2
3−2𝑥𝑥+1
j.
d.
k.
(3𝑥𝑥 2−𝑧𝑧 ) × (𝑥𝑥 4 ) ÷ (32 𝑥𝑥 −4 )
l.
�
e.
f.
m.
g.
h.
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16𝑦𝑦 7 𝑧𝑧 3
20 𝑦𝑦 −2 𝑧𝑧 8
Page 23 of 31
3𝑥𝑥 2 8𝑥𝑥 −3 𝑦𝑦 4
� � 3 −5 �
−2𝑦𝑦
3 𝑥𝑥
−2𝑥𝑥 2 𝑦𝑦 15𝑥𝑥 0 𝑦𝑦
×
5𝑦𝑦 −3
𝑥𝑥𝑦𝑦 −2
Math 10
1.10 EXPONENT APPLICATIONS
1. Solve the following equations for x.
c. 3𝑥𝑥 = 27
f. 62𝑥𝑥−2 = 6𝑥𝑥+3
a. 5𝑥𝑥 = 56
d. 81 = 9𝑥𝑥
1 3
g. � 𝑥𝑥 � = 4
4
b. 𝑛𝑛3 = 𝑛𝑛 𝑥𝑥
𝑥𝑥+3
Ex. 105𝑥𝑥+2 = 1012
𝟓𝟓𝟓𝟓 + 𝟐𝟐 = 𝟏𝟏𝟏𝟏
𝟓𝟓𝟓𝟓 = 𝟏𝟏𝟏𝟏
𝒙𝒙 = 𝟐𝟐
e. 8
h. 216 = 64𝑥𝑥+5
7
=8
2. Solve. Round your answers to the nearest hundredth. Reminder: The opposite of a
power is a root.
Ex. 66 = 𝑥𝑥 3
d. 𝑑𝑑2 − 9 = 254
𝟑𝟑
√𝟔𝟔𝟔𝟔 = 𝒙𝒙
𝒙𝒙 = 𝟒𝟒. 𝟎𝟎𝟎𝟎
a. 37 = 𝑥𝑥
e. ℎ3 + 100 = 725
2
f.
b. 102 = 𝑦𝑦 3
4
= 79
g. 1038 = 6𝑥𝑥 3
c. 𝑎𝑎2 = 550
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𝑚𝑚2
Page 24 of 31
Math 10
3. Find the
length of a side for each square. Round your answers to the nearest
hundredth. Reminder: The formula for the area of a square is 𝐴𝐴 = 𝑠𝑠𝑠𝑠𝑠𝑠𝑒𝑒 2
a. A square with area: 700 𝑐𝑐𝑐𝑐2
c. A square with area: 15 𝑚𝑚𝑚𝑚2
b. A
d. A square with area: 2700 𝑚𝑚2
square
with
area:
48.5 𝑘𝑘𝑘𝑘2
4. Find the radius of each circle. Use 3.14 for 𝜋𝜋. Round your answers to the nearest
thousandth. Reminder: The formula for the area of a circle is 𝐴𝐴 = 𝜋𝜋𝑟𝑟 2
a. A circle with area: 29 𝑘𝑘𝑚𝑚2
c. A circle with area: 504 𝑐𝑐𝑚𝑚2
b. A circle with area: 221 𝑚𝑚2
d. A circle with area: 8060 𝑚𝑚2
5. Find the radius of each sphere. Use 3.14 for 𝜋𝜋. Round your answers to the nearest tenth.
4
Reminder: The formula for the volume of a sphere is 𝑉𝑉 = 3 𝜋𝜋𝑟𝑟 3
a. A sphere with volume: 29 𝑚𝑚𝑚𝑚3
c. A sphere with volume: 950 𝑐𝑐𝑚𝑚3
b. A sphere with volume: 110 𝑚𝑚3
d. A sphere with volume: 5 𝑘𝑘𝑚𝑚3
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Math 10
6. You have $10 000 in a Education Savings Fund. It is in an investment with a
compounding annual interest rate of 3%. What is the value of the fund after 15 years?
Use the following formula: 𝐴𝐴 = 𝑃𝑃(1 + 𝑖𝑖)𝑛𝑛
A = Final value (Principle + Interest)
P = Principal amount (Starting or Initial investment amount)
i = interest rate as a decimal
n = number of years invested
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Math 10
UNIT 1 – ANSWER KEY
SECTION 1.1
1. a. 74 b. 93 c. 41 d. 510 e. 202 f. 23 g. 126 h. 452 i. 78
2. a. 2 × 2 × 2 × 2 b. 7 × 7 c. 9 × 9 × 9 × 9 × 9 × 9 × 9 × 9
3×3
3. a. 256 b. 3125 c. 729 d. 390 625 e. 81
4. a. = b. > c. < d. = e. < f. > g. < h. =
5.
Perfect Squares
Exponent
Expanded
12
22
32
42
52
62
72
82
92
102
112
122
1×1
2×2
3×3
4×4
5×5
6×6
7×7
8×8
9×9
10 × 10
11 × 11
12 × 12
Perfect
Square
1
4
9
16
25
36
49
64
81
100
121
144
6. 302 , 400, 64, 112 , 625
SECTION 1.2
1. a. 73 b. 196 c. 36 d. -2 e. 25
2. a. -311 b. -19
3. a. 51 b. 9 c. -3 d. -316
4. a. 10 b. 6 c. 30 d. -128 e. 69 f. 70
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d. 25 e. 3 × 3 × 3 ×
Math 10
SECTION 1.3
1. a. 7 b. 10 c. 2 d. 12 e. 9
2. a. √9 = 3 b. √64 = 8 c. 82 = 64 d. 122 = 144 e. √196 = 14
3. a. 4 b. 8 c. 9 d. 1 e. 12 f. 6 g. 10 h. 11
4. a. 16 b. 22 c. 35
5. a. No b. No c. Yes, 25 d. No e. Yes, 19 f. Yes, 27
6. Because the factor pair of a perfect square is a repeat of itself, therefore you only see
one of that number in the list of factors, turning the list of factors pairs into a odd
number.
SECTION 1.4
1.
2. a. 7 & 8 b. 4 & 5 c. 1 & 2 d. 5 & 6 e. 11 & 12
3. a. √4 & √9 b. √49 & √64 c. √81 & √100 d. √16 & √25 e. √1 & √4
4. a. ≈ 6.8 b. ≈ 4.6 c. ≈ 11.2 d. ≈ 5.4
5. a. 6.78 b. 4.58 c. 11.22
6. a. 1.41 b. 9.8 c. 22.36 d. 8.66 e. 141.42
SECTION 1.5
1. a. Yes b. Yes c. No d. No e. Yes
2. Note: a & b can be switched places, but c must always be the longest side.
3. 2, 5, 3, 1, 4
4. a. 10 cm b. 13 cm c. 35 mm d. 20 cm e. 26 m
5. a. 6.08 cm b. 17.80 km c. 78.87 m
6. a. 15 cm b. 13.2 m
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Math 10
7. 6.a. Because all of its sides are positive integers.
8. a. Yes b. No c. Yes
9. a. 18.4 in
b. Yes, the corners are square. This is because the triangles formed by the
diagonal are right triangles (as proven by Pythagoras’ Theorem), which means the
angles in the corners of the door jamb are 90°. c. 200.3 km
SECTION 1.6
1. 303 , 8, 216, 64, 1000, 125
2.
Perfect Cubes
Exponent
Expanded
13
23
33
43
53
63
103
1×1×1
2×2×2
3×3×3
4×4×4
5×5×5
6×6×6
10 × 10 × 10
Perfect
Cube
1
8
27
64
125
216
1000
3. a. 4 b. 9 c. 7 d. 8
3
3
3
3
3
3
3
3
3
3
4. √1, √8, √27, √64, √125, √216, √343, √512, √729, √1000
5. Note: Answers +/- 0.1 from the stated answer are considered accurate estimations.
a. ≈ 3.6 b. ≈ 5.2 c. ≈ 1.8 d. ≈ 4.9 e. ≈ 9.8 f. ≈ 1.3 g. ≈ 6.5 h. ≈ 8.1
6. a. 7.37 b. 2.52 c. 4.64 d. 10.32 e. 1.26 f. 25 g. 3.3 h. 50
SECTION 1.7
1. a. 76 b. 29 c. 55
1 7
2. a. 89 b. 312 c. 94 d. 158 e. �3�
3. a. 101 𝑜𝑜𝑜𝑜 10 b. 64 c. 140
4. a. 253 b. 64 c. 3−4 d. 72 e. 41 𝑜𝑜𝑜𝑜 4
5. c, a, e, h, b, g, f, d
6. a. 𝑥𝑥13 b. 𝑛𝑛11 c. 𝑎𝑎4 d. 𝑦𝑦 4 e. 𝑚𝑚5 f. 𝑓𝑓 5
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Math 10
7. a. 𝑥𝑥 7 𝑦𝑦10 b. 𝑎𝑎9 𝑏𝑏 5 c. 𝑥𝑥 7 𝑦𝑦 7 d. 𝑦𝑦 3 𝑧𝑧 e. 𝑟𝑟 3 𝑠𝑠 4 𝑡𝑡 2 f. 𝑎𝑎4 𝑏𝑏𝑐𝑐 2
ℎ𝑤𝑤 4
8. a. 27𝑥𝑥 3 𝑦𝑦 2 b. 𝑑𝑑 5 𝑥𝑥 3 𝑓𝑓 c. 25𝑑𝑑
SECTION 1.8
1. a. 𝑥𝑥 20
d. 32𝑎𝑎2 𝑏𝑏 e. 𝑥𝑥 7 𝑦𝑦 2 𝑧𝑧 3 f. 2𝑚𝑚4 𝑛𝑛8
b. 𝑎𝑎10 𝑏𝑏15 c. 𝑑𝑑 24 𝑒𝑒 8 𝑓𝑓 16 d. 64𝑛𝑛18 e. 25𝑥𝑥14 f. 27𝑎𝑎6 𝑏𝑏 3 g. 256𝑓𝑓 8 𝑔𝑔4 ℎ40
2. a, b, e, h, i
3. a. -8 b. -16 c. 16 d. −32𝑥𝑥 5 e. 64𝑥𝑥 6
4. a. NEG b. POS c. NEG d. NEG e. NEG f. NEG
g. POS h. NEG
5. a. 𝑑𝑑17 𝑒𝑒 19 b. 𝑥𝑥 28 𝑦𝑦13 𝑧𝑧10 c. 125𝑚𝑚20 𝑛𝑛6 d. −432𝑎𝑎17 𝑏𝑏 8
1
1
1
1
𝑦𝑦 2
5
6. a. 46 b. 𝑦𝑦 3 c. 54 𝑥𝑥 4 𝑜𝑜𝑜𝑜 (5𝑥𝑥)4 d. 𝑥𝑥 4 e. 38
22
𝑦𝑦 8
7. a. 32 b. 𝑥𝑥 2 c. 5𝑦𝑦 6 d. 34 e. 𝑥𝑥 2 f. (32 )(33 ) 𝑜𝑜𝑜𝑜 35
1
23
1
8. a. 46 b. 46 c. 𝑥𝑥 0 d. 170 e. 9981 f. 𝑦𝑦 6
9. a. (−4𝑡𝑡 + 5)(−4𝑡𝑡 + 5) b. (𝑥𝑥 − 3)(𝑥𝑥 − 3)(𝑥𝑥 − 3)(𝑥𝑥 − 3)
c. (3𝑎𝑎 + 2𝑏𝑏 − 6)(3𝑎𝑎 + 2𝑏𝑏 − 6)(3𝑎𝑎 + 2𝑏𝑏 − 6) d. (6𝑚𝑚2 𝑛𝑛)(6𝑚𝑚2 𝑛𝑛)(6𝑚𝑚2 𝑛𝑛)(6𝑚𝑚2 𝑛𝑛)
e. (−2𝑥𝑥 2 + 5𝑥𝑥 + 4)(−2𝑥𝑥 2 + 5𝑥𝑥 + 4)
1
10. a. 𝑎𝑎6 b. 81𝑦𝑦 2
SECTION 1.9
1. a. 5 b. 3v c. 6ab d. x e. 8
2. a. -4 b. 14 c. 345 d. none e. 1/10
16
1
243𝑎𝑎20
𝑤𝑤 3
𝑦𝑦 3
𝑧𝑧 8
3. a. 𝑦𝑦 2 b. 𝑔𝑔6 ℎ6 c.
𝑏𝑏 10
64𝑎𝑎12
e. −8𝑥𝑥 3 𝑦𝑦 3
d. 𝑏𝑏12
1
64
𝑥𝑥 8 𝑧𝑧 8
4096
16𝑦𝑦 9
4. a. 125 b. 3𝑥𝑥 4 c. 25ℎ4 d. 216𝑓𝑓15 e. 81𝑦𝑦 8 𝑧𝑧 2 f. 16𝑦𝑦 4 g. 𝑔𝑔2 𝑘𝑘 10 𝑧𝑧 6 h. 𝑧𝑧 5 i. 2𝑥𝑥 7+𝑥𝑥
33𝑥𝑥
j. 33x-3 𝑜𝑜𝑜𝑜 27
2020-11-05
𝑥𝑥 10−𝑧𝑧
k.
3
l. −
4𝑥𝑥 4 𝑦𝑦 3
9
m. −6𝑥𝑥𝑦𝑦 7
Page 30 of 31
Math 10
SECTION 1.10
1
1
1. a. 6 b. 3 c. 3 d. 2 e. 4 f. 5 g. − 3 h. − 2
2. a. ±6.08 b. 4.67 c. ±23.45 d. ±16.22 e. 8.55 f. ±17.78 g. 5.57
3. a. 26.46 cm b. 6.96 km c. 3.87 mm d. 51.96 m
4. a. 3.039 km b. 8.389 m c. 12.669 cm d. 50.664 m
5. a. 1.9 mm b. 3.0 m c. 6.1 cm d. 1.1 km
6. $15 579.67
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