Explanation of Multiplication with Exponents - Succeed in Understanding Algebra. Also refer to number, variable, raised to a power, mathematics, math, maths, Ron Kurtus, School for Champions. Copyright © Restrictions
Multiplication with Exponents
by Ron Kurtus (revised 10 January 2009)
When you multiply two numbers or variables with 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 break it up into several multiplications.
Questions you may have include:
- How do you multiply numbers raised to a power?
- How do you multiply variables raised to a power?
- How do you include other numbers when multiplying?
This lesson will answer those questions. There is a mini-quiz near the end of the lesson.
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.
Same 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) =
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.
Different base
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.
Multiplying variables
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.
Same base
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) =
x*x*x*x*x*x*x = x7
Also, when both the base and exponents are variables, (xa)*(xb) = xa+b.
Different base
If the base numbers are different, this rule does not apply. For example (x6)*(y6) 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,
(12*105)*(2*103) =
(12*2)*(105*103) =
24*108 =
2.4*10*108 = 24*109
Note that you change 24*108 to 2.4*109 because the standard convention is that the first number in raising 10 to a power should be between 1 and 10. Since 24 is more than 10, you can divide 24 by 10 and multiply 108 by 10 to get 2.4*109.
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) = 7*y2*x8
(a3*b3)*(b6*a5) = a8*b9
Summary
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.
See side menu for more Algebra topics
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Resources
The following resources provide information on this subject:
Websites
Books
Mini-quiz to check your understanding
If you got all three correct, you are on your way to becoming a Champion in Algebra. If you had problems, you had better look over the material again.
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Multiplication with Exponents