1. Using the product rule
2. Evaluating expressions raised to the zero power
3. Using the quotient rule
4. Evaluating expressions raised to the negative powers
5. Raising a power to a power
6. Converting between scientific notation and standard notation
Exponents may be used to write repeated factors in an expression.
32 3 3 In this case 3 is the base and 2 is the power or exponent.
This means we can say that 3 to the second power means that we are
multiplying 3 by itself 2 times.
y3 y y y In this case y is the base and 3 is the exponent.
This means that y is raised to the third power or y is multiplied by itself 3
Both of these are examples of exponential expressions. These are
expressions that can be multiplied, divided, added, and subtracted or raised
to another power. In this module we look at multiplying, dividing and
raising to powers.
a 2 a 3 a a a a a a 23 a 5
This suggests the following rule of multiplying like bases.
Product Rule for Exponents
If m and n are positive exponents and a is a real number then,
a m a n a m n
In other words, when we multiply like bases we add the exponents.
Example 1: Multiplying like bases
a. 23 24 b. b 4b5 c. y 2 y y3
a. 23 24 234 27 b. b 4b5 b 45 b9
c. y 2 y y3 y213 b6
Example 2: Using the product rule and properties.
a. 3 x 2 4 x b. 2a 3b 5ab 4
a. 3 x 2 4 x 3 4 x 2 x 12 x 21 12 x 3
b. 2a 3b 5ab 4 2 5a 3 a b b 4 10 a 31b1 4 10 a 4b5
We define the 0 power to be 1 in order that all the rules for exponents
remain true if the exponent is zero.
If a does not equal 0 then a 0 1
Example 3: Zero exponents
b. 50 d. x y
a. 50 c. 2x 0
a. 50 1
b. 50 1
c. 2 x 0 2 1 2
d. x y = 1
We now look at dividing like bases keeping in mind that division is the
opposite of multiplication. Since when we multiplied like bases we added
the exponents, dividing like bases would indicate that we should subtract the
x5 x x x x x
x 5 3 x 2
From this we define division as the following.
Quotient Rule for Exponents
If a is a nonzero real number and n and m are integers, then
In other words, when dividing like bases subtract the exponents
Example 4: Using the quotient rule
x7 15 x 3 y 6
a. 5 b.
x 3x 2 y 4
a. 5 x 75 x 2
15 x3 y 6
b. 2 4
5 x32 y 64 5 xy2
When the exponent in the denominator is larger than the exponent of the
numerator, applying the quotient rule will result in a negative exponent. For
x57 x 2 But we could also evaluate it as follows.
x5 x x x x x 1
x7 x x x x x x x x
From this we see that
x 2 2
In general we then define negative exponents to be the reciprocal of the base
raised to the positive exponent.
If a is a real number other than 0 and n is a positive integer, then
a n n
Example 5: Evaluating Expressions with Negative exponents
a. 32 b. 2 x 3
1 1 2 1
a. 32 2 b. 2 x 3
3 9 x3 3a
Raising a Power to a Power
Sometimes it is necessary to raise exponential expressions themselves to a
power. For example
x2 3 x2 x2 x2 x222 x6 .
Notice that if we had multiplied the exponents the result is the same
x2 3 x23 x6
Power Rule for Exponents
If m and n are positive integers and a is a real number , then
am n amn amn
Remember when raising a power to a power, keep the base the same and
multiply the exponents.
Example 6: Raising powers to powers
a. b3 b. 2x 4
a. b3 b35 b15
b. 2 x4 2 x43 8x12
Very large and very small numbers occur frequently in nature. For example
the distance from Earth to the Sun is very large whereas the diameter of a
helium atom is very small. Some calculators do not operate with very large
or very small numbers, so scientific notation is a shorthand way of writing
these numbers so all calculators can handle them.
A positive number is written in scientific notation if it is written as the
product of a number a, where 1 a 10 and an integer power of b of 10:
The following are examples of numbers written in scientific notation.
To evaluate 2.3 104 means that we move the decimal place 4 places to the
right. Thus, 2.3 10 4 23,000 .
To evaluate 1.6 103 means that we move the decimal 3 places to the left.
Therefore, 1.6 103 0.0016
Similarly to write a number in scientific notation we reverse the process.
120,000 1.2 105
0.000234 2.34 104
Writing a Number in Scientific Notation
1. Move the decimal point in the original number until the new
number has a value between 1 and 10.
2. Count the number of decimal places the decimal point was
moved in Step 1. If the decimal point was moved to the left, the
count is positive. If the decimal point was moved to the right,
the count is negative.
3. Write the product of the new number in Step 1 and 10 raised to
an exponent equal to the count found in Step 2.
Example 7: Changing a number to scientific notation
Write each number in scientific notation
a. 63,000,000 b 0.000017
a. 63,000 ,000 6.3 10 7
b. 0.000017 1.7 105
Writing a Number in Standard Notation from Scientific Notation
Move the decimal point in the number the same number of places as
the exponent on 10. If the exponent is positive, move the decimal
point to the right. If the exponent is negative, move the decimal
point to the left.
Example 7: Changing to Standard Notation
Write each number in standard notation
a. 5.24 106 b 1.98 103
a. 5.24 10 6 5,240 ,000 . Notice the decimal point has been moved 6
places to the right.
b. 1.98 103 0.00198 Notice that the decimal has been moved 3 places to
Many calculators now perform calculations using scientific notation. Check
in the user manual with your calculator to see if it converts and uses