Download Exponents and Radical Review

EXPONENTS AND RADICAL
REVIEW
Prep for Chapter 10
Exponents and Radicals Review
I. Inverse Operations Review
OPERATION
INVERSE
Addition
Subtraction
Subtraction
Addition
Multiplication
Division
Division
Multiplication
Square Root
Squaring
Squaring
Square Root
Finding a root of a number is the inverse operation of raising
a number to a power. (You can use other roots and powers
besides 2).
Exponents and Radicals Review
II. Exponents
The exponential form of a number is below.
The exponent tells us how many times the base will be
multiplied by itself.
The base is the number being multiplied.
A number produced by raising a base to an exponent
is called a power. 27 and 33 are equivalent.
Exponent
Base
2
7
Exponents and Radicals Review
Examples:
Write in exponential form.
A. 4 • 4 • 4 • 4• 4
4•4•4•4=
45
Identify how many times 4
is a factor.
B. (–6) • (–6) • (–6)
(–6) • (–6) • (–6) = (–6)3
Identify how many times –6
is a factor.
Reading Math
Read (–6)3 as “–6 to the 3rd power" or "–6 cubed”.
Exponents and Radicals Review
III. Radical Expressions
A. Terminology and Symbols
This symbol is the radical or the radical sign
radical sign
index
n
a
radicand
The expression under the radical sign is the radicand.
The index defines the root to be taken.
Exponents and Radicals Review
IV. Square Roots
A square root of any positive number has two roots – one is
positive and the other is negative. For example, the square
root of 64 is both 8 and -8.
When simplifying square roots (as opposed to solving them),
we only use the positive, or principal, root.

The symbol
The above symbol
represents the
represents the
negative root of a
positive or principal number.
root of a number.
Ex:
64
simplified
Ex: 64 simplified
is -8
is 8
−
The square root
of a negative
number is a nonreal number.
Why?
Exponents and Radicals Review
To summarize,
A negative outside the radical of a square root means you
take the negative of the root.
A negative inside the radical of a square root means it is a
non-real number.
Examples:
100  10
 36  6
9  non-real #
0.81  0.9
Exponents and
Rdicals
Radicals Review
V. Cube Roots
3
a
A cubed root is asking “What base is multiplied three times
to get the radicand?” In other words, “What times itself, times
itself again, give you the radicand?
Examples:
3
27  3
3
8  2
A cube root of any positive number is positive.
A cube root of any negative number is negative.
Exponents and Radicals Review
VI. nth Roots
An nth root of any number a is a number whose nth power is a.
Examples:
34  81
4
81  3
2  16
4
16  2
5
32  2
4
 2 
5
 32
Exponents and Radicals Review
nth Roots
Be careful with negatives inside the radical.
Examples:
5
1  1
4
16  Non-real number
6
1  Non-real number
3
27  3
Do you see a pattern?
If there is a negative
under the radical and
the index is positive, it
is a non-real number.
If there is a negative
under the radical and
the index is negative,
the root will be
negative.
VII. PRIME FACTORIZATION
Breaking down a number into its prime
factorization will help you simplify radical
expressions in 10-1.
A. PRIME AND COMPOSITE NUMBERS
A whole number, greater than 1, for which the
only factors are 1 and itself is called a prime
number.
Examples: 2, 3, 5, 7, 11, 13, 17, 19
A whole number, greater than 1, that has more than two
factors is called a composite number.
Examples: 4, 6, 8, 9, 10, 12, 14, 15, 20
0 and 1 are NEITHER prime nor composite.
B. Finding the prime factorization of a number
A whole number expressed as the product of prime numbers is called the prime
factorization of the number.
Ex. 1: Find the prime factorization of 90
90
Make a factor tree
What are some
factors of 90?
Can we break down
9 and 10 further?
Write factors for
each number that
is composite.
Circle the prime
numbers.
9
3
10
35
2
90 = 2 ∙ 3 ∙ 3 ∙ 5 𝑜𝑟 90 = 2 ∙ 32 ∙ 5
Notice how the numbers are in order from least to
greatest.