Hello. I'm Professor Von Schmohawk and welcome to Why U.
So far we have studied the properties of the four basic arithmetic operations
addition, subtraction, multiplication, and division.
In this lecture, we will introduce the mathematical operation of exponentiation.
We saw that multiplication is just shorthand for repeated addition.
For example, multiplying 4 times 10
is the same as adding four 10's together.
Let's state this in a more general way.
If we let the letter "n" represent any integer and "a" represent any number
we can say that "n" times "a" is the same as adding n a's together.
Just as multiplication is shorthand for repeated addition
exponentiation is shorthand for repeated multiplication.
"a" raised to the nth power
is the same as multiplying n a's together.
To write "a" raised to the nth power
we write the number "a" which is called the "base"
with a small superscript "n" which is called the "exponent".
This forms an "exponential expression".
For example, the exponential expression "4 raised to the second power"
is written as the number 4 with the exponent 2 written as a superscript.
Since raising a number to the second power
is equivalent to multiplying two copies of that number together
which is the same operation one would perform when calculating the area of a square
the number is said to be "squared".
Likewise, raising a number to the third power
is equivalent to multiplying three copies of that number together.
Since this is the same operation one would perform when calculating the volume of a cube
the number is said to be "cubed".
Individual exponential expressions can be combined into larger expressions
through operations such as addition, subtraction, multiplication and division.
For instance, we can multiply the two exponential expressions "4 squared" and "4 cubed"
creating a larger expression.
We sometimes refer to these individual elements within a larger mathematical expression
as "terms".
Interesting things happen when we multiply exponential expressions together.
For example, say that we multiply "4 squared" times "4 cubed".
Since "4 squared" is 4 times 4
and "4 cubed" is 4 times 4 times 4
the result is five 4's multiplied together
which is equivalent to 4 to the fifth power.
So 4 to the second power times 4 to the third power is 4 to the fifth power.
Notice that we can get this result by simply adding the exponents.
This is true any time we multiply exponential expressions with the same base.
For example, multiplying 5 to the third power times 5 to the fourth power
is the same as multiplying seven 5's
which is 5 to the seventh power.
Once again, we can get this result by adding the exponents.
So to multiply exponential expressions with the same base, add their exponents.
But what happens if we divide two exponential expressions with the same base?
Let's try as an example, 5 to the seventh power divided by 5 to the fourth power.
This is the same as seven 5's multiplied together
divided by four 5's multiplied together.
We can simplify this fraction by canceling four 5's in the numerator
and four 5's in the denominator.
After canceling, we are left with three 5's multiplied together.
So 5 to the seventh power divided by 5 to the fourth power is 5 to the third power.
Notice that we can get this result
by subtracting the exponent of the denominator from the exponent of the numerator.
So to multiply exponential expressions with the same base, add their exponents
and to divide exponential expressions with the same base, subtract their exponents.
Let's summarize these rules of exponents by using letters instead of specific numbers.
To multiply exponential expressions with the same base, add their exponents
and to divide exponential expressions with the same base, subtract their exponents.
In the next lecture we will use these rules to explore more properties of exponents
including the meaning of exponents of one, zero, or any negative integer.