Multiplication with Exponents
by Ron Kurtus (revised 13 November 2015)
When you multiply exponential expressions that have the same base, you simply add the exponents. This is true for both numbers and variables. When you include other numbers in the multiplication, you simply factor and break the expression into several multiplications.
Note: The base of the exponential expression xy is x and the exponent is y.
This rule does not apply when the numbers or variables have a different base.
Questions you may have include:
- How do you multiply exponential numbers?
- How do you multiply variables raised to a power?
- How do you include other numbers when multiplying?
This lesson will answer those questions.
Multiplying exponential numbers
When you multiply two numbers or variables with the same base, you add the exponents. This rule does not hold if the numbers are of a different base.
A demonstration or verification of that rule is seen when you multiply 7*7*7 times 7*7. The result is:
7*7*7*7*7 = 75
But 7*7*7 = 73 and 7*7 = 72. Thus, 73*72= 73+2 = 75.
Also, 23*25*22 = 23+5+2 = 210.
You can see that when you multiply numbers of the base raised to a power, you add their exponents.
When rule does not apply
This rule does not apply when multiplying exponents of a different base.
For example, you cannot add exponents in 32*42. The numbers must be multiplied out as 32*42 = 9*16.
When you multiply two variables with the same base, you add the exponents. You cannot do that when the bases of the exponential numbers are different.
Thus x3*x4 = x3+4 = x7. This can be proved, since x3 = x*x*x and x4 = x*x*x*x, then
x*x*x*x*x*x*x = x7
Also, when both the base and exponents are variables, (xa)*(xb) = xa+b.
If the base numbers are different, this rule does not apply. For example (x6)*(y3) cannot be simplified.
Including other numbers
If you have exponential numbers that are multiplied by other numbers, you can easily do the arithmetic. For example, simplify:
Rearrange the numbers:
Then add the exponents:
The other numbers or variables can also be exponentials. Some examples include:
(33*52)*(53*33) = (33+3)*(52+3) = 36*55
(7*x3)*(y2*x5) = 7y2x8
(a3*b3)*(b6*a5) = a8b9
When you multiply two numbers or variables with the same base, you simply add the exponents. This is true for both numbers and variables.
For example, 23*27= 210 and c3*c4 = c7.
When you include other numbers or variables in the multiplication, you simply break it up into several multiplications, such as (x*105)*(x*103) = x2*108.
Always do your best
Resources and references
Exponents: Basic Rules - PurpleMath.com
Exponent Rules - RapidTables.com
Exponents Calculator - CalculatorSoup.com
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