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.

Thank you sir!

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

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

Thank you so much!!

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

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

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.

Nice!

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