In this lesson, our instructor Vincent Selhorst-Jones will help you understand exponents. Youll receive an introduction to both the fundamental idea and the expanded idea. Youll also learn how to handle exponents that have a power to the zero, to the negative, to the irrational, and to the fraction. Vincent ends the lesson with a complete summary of rules and then five fully worked-out examples.
At heart, exponentiation is repeated multiplication. By definition, for any number x and any positive integer a,
x·x ·x …x ·x
We can expand on this fundamental idea to see how exponentiation can work with numbers that aren't just positive integers.
Through multiplication, we can combine numbers that have the same exponent base:
xa ·xb = xa+b.
We can consider exponentiation acting upon exponentiation:
(xa)b = xa·b.
We can look at raising two numbers to the same exponent:
xa ·ya = (xy)a.
For any number at all, raising it to the 0 turns it into 1:
x0 = 1.
Raising a number to a negative will "flip" it to its reciprocal:
Because of the above, we see that a denominator is effectively a negative exponent. This means if we have a fraction where the numerator and denominator have the same base, we can subtract the denominator's exponent from the numerator's exponent:
Raising a number to a fraction is the equivalent of taking a root:
x[1/2] = √x, x[1/3] =
, x[1/n] =
If we want to find the value of raising something to an irrational number, we can find a decimal approximation of the true value by just using many decimals from our irrational number:
8π = 83.1415926… ≈ 83.14159.
The more accurate we need our approximation to be, the more decimals we can use from the irrational number.
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.