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Review Exponents
An exponent is a shortcut way to write repeated multiplication. The expression 23 is two
multiplied three times: 2:2:2 = 8. Rules for operations with exponents are based on this
fact. The parts of an exponential expression are the base (2 in the example), and the power
or exponent (3 in the example). If the exponent is zero, the expression is equal to zero.
Multiplying Exponents
When multiplying expressions with the same base, add the exponents. If a number has no
exponent, the exponent is one.
x2:x3 = (x:x):(x:x:x) = x5 2:24 = (2):(2:2:2:2) = 25 = 32
?
1. Multiply: y4:y2
Add the exponents: y4:y2 = y4+2 = y6
Dividing Exponents
When dividing expressions with the same base, subtract the exponents.
x:x:x:x
x4
24 2 : 2 : 2 : 2
2
=
=
=
=
= 2:2 = 4
x
x
x
:
x2
22
x:x
2:2
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2. Divide: y6 ÷ y5
Subtract the exponents: y6 ÷ y5 = y6-5 = y1 = y
Negative Exponents
A negative exponent means the positive exponent is the denominator of a fraction with a
numerator of one.
x -2 =
?
3.
1
x2
7 -2 =
1
1
=
72 49
Simplify: 2-3
Make the exponent the denominator of a fraction to make the exponent positive.
2 -3 =
1
1
1
=
=
23 2 : 2 : 2 8
349
Essential Math Skills
Raising Exponents to a Power
Build Your
Math Skills
To raise a base with an exponent to a power, multiply the exponents.
(x4)2 = (x:x:x:x):(x:x:x:x) = x8
(24)2 = (2:2:2:2):(2:2:2:2) = 28 = 256
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Using Order of
Operations with
Exponents
4. Simplify: (y5)3
You can remember
the order of
operations by
remembering
PEMDAS.
Multiply the exponents: (y5)3 = y5:3 = y15
Distributing Exponents
To apply an exponent to a product or quotient, apply it to each factor
or to the dividend and divisor.
(xy)2 = (x:y):(x:y) = x2y2 (4 ' 2)2 = (4:4)'(2:2) = 42'22 = 16'4 = 4
You cannot distribute an exponent across addition.
Parentheses
Exponents
(and Roots)
Multiplication
Division
Addition
Subtraction
Incorrect  (2 + 3)2 = 22 + 32 = 4+9 = 13
Correct  (2 + 3)2 = 52 = 25
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5. Simplify: (2y2)3
Raise each number or variable to the exponent: (2y2)3 = 23y6 = 8y6
Scientific Notation
Scientific notation is a handy way to use exponents to make
working with very large or very small numbers easier. A news
article about population growth or nanotechnology might use
scientific notation to talk about large and small numbers.
Scientific notation shows a decimal of the value 1 ≤ x < 10
(such as 5.25) multiplied by a power of 10 (such as 5.25 × 105).
The exponent tells you how many places right (for a positive
exponent) or left (for a negative exponent) to move the decimal to convert the number to standard form. If the exponent
is negative, the standard form is less than one. If it is positive,
the standard form is 10 or greater.
350
Standard
Form
Scientific
Notation
5.25
5.25
52.5
5.25 × 10
525,000
5.25 × 105
52,500,000
5.25 × 107
0.00525
5.25 × 10-3
0.0000525
5.25 × 10-5