# Pre-Algebra 21 – Exponentiation

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.

### 9 thoughts on “Pre-Algebra 21 – Exponentiation”

• February 20, 2012 at 12:02 pm

Thank you sir!

• April 27, 2012 at 2:48 am

I showed this video to my class today! And sorry but I caught one more mistake, at 4:38 you say "denominator" when you cross out the 5s in the numerator, and vice versa. Other than that this video was awesome and my kids enjoyed it a lot! Keep up the great work! -Justin

• March 21, 2013 at 9:04 am

this video can do which my sir and books failed to do

• January 16, 2014 at 1:58 am

Thank you so much!!

• February 15, 2014 at 9:58 pm

awsome.. I'm in my way to revisit math because my teacher when I was a kid made me hate math đ

• February 26, 2015 at 9:00 am

Like if you know all this stuff and watch this video to improve your English.

• September 8, 2015 at 1:43 pm

Isn't it incorrect to refer to the factors of 4^2*4^3 as terms? I thought that terms meant quantities taking part in either addition or subtraction … I would have named them factors.

• November 23, 2015 at 8:40 pm

Nice!

• February 26, 2016 at 5:23 pm

.