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Name _______________________________________ Date __________________ Class __________________
Exponents and Roots
SECTION A Family Letter: Exponents
Dear Family,
The student will be learning about exponents and the properties
associated with them. An exponent is a number that represents
how many times the base is to be multiplied by itself. A number
produced by raising a base to an exponent is called a power.
Using exponents will allow the student to write repeated
multiplication in a more efficient way. This is how the student
will write and evaluate exponents.
Vocabulary
These are the math
words we are learning:
scientific notation
a shorthand way of
writing very large or
very small numbers
using powers of 10
Write in exponential form.
2•2•2•2
2 • 2 • 2 • 2  24
Identify how many times 2 is a factor.
Simplify 43.
43  4 • 4 • 4
Find the product of three 4’s.
 64
Sometimes the exponent will be zero. The zero power of any
number except 0 equals 1.
When simplifying expressions with exponents, remind the
student to follow the order of operations.
There are important relationships that exist between exponents
and the operations of multiplication and division. The student
will learn how to multiply and divide powers with the same
base by using some of these basic properties of exponents.
Multiplying
When multiplying powers with the same base, keep the base
constant and add the exponents.
Multiply. Write the product as one power.
56 • 52
562
Add the exponents.
58
Dividing
When dividing powers with the same base, keep the base
constant and subtract the exponents.
Divide. Write the quotient as one power.
1214
129
1214–9
Subtract the exponents.
125
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
Exponents and Roots
SECTION A Family Letter: Exponents continued
When multiplying or dividing powers with the same base it is
important to remember not to multiply or divide the bases. This
property only applies when bases are the same. Powers with unlike
bases cannot be combined using the two properties shown in the
examples.
It is possible to raise a power to a power. To simplify a power raised
to a power, keep the base and multiply the exponents.
Since it is possible to have a sum or difference that is a negative
number, the student will learn how to evaluate expressions with
negative exponents. A number raised to a negative exponent equals
1 divided by that number raised to the opposite of the exponent. This
is how the student will learn to evaluate expressions with negative
exponents.
Divide. Write the quotient as one power.
54
56
52 The bases are the same so subtract the exponents.
1
Write the reciprocal and change the sign of the exponent.
52
1
Simplify.
25
Scientific notation is a shorthand way to write very small or very
large numbers. Numbers in scientific notation are written as the
product of a decimal and ten with an exponent.
To change numbers from scientific notation to standard notation,
move the decimal point the number of places indicated by the
exponent, to the right for a positive exponent and to the left for a
negative exponent. Similarly, to convert numbers from standard to
scientific notation, move the decimal point and multiply by ten to the
power of the number of places the decimal point moves.
Multiplying and dividing numbers in scientific notation works the
same way as with other exponents.
Have the student explain the purpose of exponents and the many
ways they are used in mathematics. Verbalizing this information
helps understanding this material.
Sincerely,
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
Exponents and Roots
SECTION A At-Home Practice: Exponents
Write using exponents.
1. (4) • (4) • (4)
2. 3 • 3 • 3 • 3 • 3 • 3
_______________
3. 9 • 9
_______________
4. t • t • t
_______________
________________
Simplify.
5. 43
6. (3)4
_______________
9. (2  43)
7. 64
_______________
10. (27 – 32)
_______________
8. (1)9
_______________
11. 25 + (8 • 43)
_______________
________________
12. (12  72)
_______________
________________
Multiply. Write the product as one power.
13. 64 • 63
14. 53 • 56
_______________
15. 82 • 87
_______________
16. k8 • k2
_______________
________________
Divide. Write the quotient as one power.
17.
56
54
18.
_______________
r7
r6
19.
_______________
812
20.
86
_______________
p8
p4
________________
Simplify the powers of 10.
21. 103
22. 106
_______________
23. 108
_______________
24. 101
_______________
________________
Simplify.
25. 33
26.
_______________
62
6
27.
5
_______________
45
4
28. 74 • 75
3
_______________
________________
Write each number in standard or scientific notation.
29. 9,630,000
_______________________
30. 2.7  105
________________________
31. 0.0015
________________________
Answers: 1. (4)3 2. 36 3. 92 4. t3 5. 64 6. 81 7. 1296 8. 1 9. 66 10. 18 11. 537 12. 37 13. 67 14. 59
1
1
15. 89 16. k10 17. 52 18. r 19. 86 20. p4 21. 0.001 22. 0.000001 23. 0.00000001 24. 0.1 25.
26.
216
27
1
27. 16 28.
29. 9.63  106 30. 0.000027 31. 1.5  103
7
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
Exponents and Roots
Family Fun: Match It Up!
Directions
Using the numbers in the right-hand column, make the number
sentences true in the left-hand column. You may use some numbers
more than one time and some numbers not at all. See how well you
can match up the correct answers to the number sentences.
Number Sentences
1. 3x 
x equals ______________
Possible Answers for x
1
9
2. 7.2  10x  0.072
0
10
5
3. x12  1
2
4. 8x  83  82
7
5. x4  10,000
3
6.
78
7x
 49
7. 0.001  10x
4
1
5
8. 5x  625
3
9. 12  24 • x1
2
10. 6x • 63  36
6
11.
12.
153
15 x
1
1
 10x
10,000
4
1
10
Answers: 1. 2 2. 2 3. 1 4. 5 5. 10 6. 6 7. 3 8. 4 9. 2 10. 5 11. 3 12.
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
Exponents and Roots
SECTION B Family Letter: Roots
Dear Family,
Vocabulary
The student will be learning about squares and square roots.
Although every positive number has two square roots, most of
the time you only write the non-negative square root, better
known as the principal square root.
These are the math
words we are learning:
Density Property
a property that states
that between any two
real numbers there is
another real number.
This property is also
true for rational
numbers, but not for
whole numbers and
integers.
hypotenuse the side
on a right triangle
across from the right
angle
irrational number
number written as a
decimal that is not
terminating or repeating
leg one of the two
sides that make up the
right angle in a right
triangle
perfect square a
number that has
integers as its square
root
principal square root
the non-negative square
root that appears on the
calculator
Pythagorean Theorem
in any right triangle, the
sum of the squares of
the legs is equal to the
square of the length of
the hypotenuse
real numbers the set
of numbers that
consists of the set of
rational numbers and
the set of irrational
numbers
The student will also learn to recognize a perfect square and
use this knowledge to evaluate expressions and estimate the
square roots of numbers that are not perfect squares.
Find the two square roots of 81.
81  9
 81  9
9 is a solution, since 9 • 9  81.
9 is also a solution since
9 • 9  81.
Remember to follow the order of operations when simplifying
expressions with exponents.
Simplify the expression, 4 64  9.
4 64  9
4(8)  9
32  9
41
Simplify the square root.
Multiply.
Add.
You can use the square roots of perfect squares to estimate
the square roots of other numbers.
The square root of 55 is between two integers.
Name the integers. Explain your answer.
55
72  49
82  64
7
Think: What perfect squares
are close to 55?
49  55
64  55
55  8
55 is between 7 and 8 because 55 is between 49 and 64.
Some square roots can be written as decimals that terminate
or repeat. Other square roots, when written as a decimal, go
on forever without repeating or terminating. These numbers
are called irrational numbers.
There is one last set of numbers that encompasses all the
sets of numbers the student has studied. The set of real
numbers includes the set of whole numbers, integers, rational
numbers, and irrational numbers. However, there are some
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
Exponents and Roots
SECTION B Family Letter: Roots continued
numbers that are not real numbers, like the square root of a
negative number. The student will learn to identify the set or
sets of numbers to which a given number belongs.
The student will also study a special kind of triangle called a
right triangle. Right triangles have one right, or 90º, angle.
The side opposite the right angle, called the hypotenuse, is
the longest side. The shorter sides, that form the right angle,
are called legs.
square root a number
that when multiplied by
itself to form a product
is the square root of
that product.
The student will learn an important formula called the
Pythagorean Theorem. This theorem relates the length of the
legs of a right triangle to the length of the hypotenuse.
The Pythagorean Theorem states that the sum of the
squares of the lengths of the two legs is equal to the square of
the length of the hypotenuse. Sometimes, the formula is a
little easier to understand.
a2  b2  c2
  
legs hypotenuse
Find the length of the hypotenuse.
a2  b2  c2
52  122  c2
25  144  c2
169  c2
169  c 2
13  c
Pythagorean Theorem
Substitute values.
Simplify powers.
Find the square root.
The Pythagorean Theorem has many useful applications. One
application that the student will learn is finding distance on the
coordinate plane. The distance between two points (x1, y1) and
(x2, y2) is
(x2  x1)2  (y 2  y1)2 . This is called the Distance
Formula, and it derives directly from the Pythagorean Theorem.
Powers, roots, real numbers, and the Pythagorean Theorem
are all ideas at the foundation of the student math education.
Understanding them is essential to his or her progress.
Sincerely,
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
Exponents and Roots
SECTION B At-Home Practice: Roots
Find the two square roots of each number.
1. 36
2. 121
_______________
3. 64
_______________
4. 900
_______________
________________
Simplify each expression.
25  4
5.
6.
_______________________
81  9
49  5
7.
________________________
________________________
Each square root is between two integers. Name the integers.
11
8.
9.  72
_______________________
10.
________________________
60
________________________
Use a calculator to find each value. Round to the nearest tenth.
72
11.
12.
_______________________
34.5
13.
________________________
824
________________________
Find a real number between each pair of numbers.
14. 4
2
3
and 4
7
7
_______________________
15.
9 and 5
16.
________________________
1
and 0
121
________________________
Find the length of the hypotenuse in each triangle.
17.
18.
_______________________________________
________________________________________
Find the distance between each pair of points. Round to the nearest tenth.
19. (2, 3) and (4, 1)
_______________________________________
20. (4, 2) and (3, 1)
________________________________________
Answers: 1. 6, 6 2. 11, 11 3. 8, 8 4. 30, 30 5. 1 6. 0 7. 12 8. 3 and 4 9. 9 and 8 10. 8 and 7 11. 8.5
5
1
12. 5.9 13. 28.7 14. 4
15. 4 16. 
17. c  34 18. c  19.8 19. 2 2 20. 7.6
14
200
Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
Exponents and Roots
Family Fun: Be a Square
Directions
Cut out the cards and shuffle them. Randomly place the cards face
down. Taking turns with your partner, choose two cards and try and
match the square root to the perfect square. When you find a pair,
keep the cards. If you do not have a match, return each card face
down in its original spot. The player with the most matches wins.
4
9
16
25
36
49
64
81
2
3
4
5
6
7
8
9
Holt McDougal Mathematics