Hello.
I’m Professor Von Schmohawk and welcome to Why U.
When we first introduced the concept of exponents
we started by defining integer exponents of two or greater
as the number of copies of the base which are multiplied together.
We then used the rules we derived for combining exponents
to figure out what the meaning of an exponent of one or zero would be
as well as the meaning of negative integer exponents.
It is natural to wonder if there would be meaning to an exponent that was not an integer.
For instance, what would be the meaning of an exponent of one-half?
We can use the rules of exponents to figure this out.
If we square "a" to the one-half, we multiply the exponents one-half and two.
This gives us "a" to the first power which is the same as "a".
So "a" to the one-half, is the number that when squared, is equal to "a".
This is the definition of a square root.
Therefore "a" to the one-half must be the square root of "a".
The idea of square roots has been around for a long time.
Both the ancient Babylonians and the ancient Egyptians knew about square roots.
If you want to construct a square object with a particular area
you need to know what length to make the sides.
If we call the length of a side x and the area of the square "a"
we know that the length, x when multiplied by itself is equal to "a".
So the problem is to find what value of x, when squared, gives us "a".
Let’s say for example, that we wanted an area of four square meters.
Then by trial and error, we could figure out that the side of the square must be two meters.
We say that two is the square root of four
since when we square the number two we get four.
But two is not the only square root of four.
Negative two is a square root of four as well
since squaring negative two also gives us four.
So four has two square roots: two and negative two.
However, a length can’t be a negative number.
Therefore we call the positive root, the principal square root.
We indicate the principal square root of four, by writing four under a radical sign.
If we want to indicate the negative square root
we place a negative sign in front of the radical
indicating the negative of the principal square root.
And if we wish to indicate both square roots
we place a plus-minus sign in front of the radical.
Any positive number has two square roots
the principal square root which is always positive
and the negative of the principal square root.
For example, nine has square roots three and negative three.
Numbers like nine and four, whose square roots are integers, are called perfect squares.
However, the square roots of most numbers don’t come out even.
In fact, most square roots have an infinite number of non-repeating digits
after the decimal point.
These kinds of numbers cannot be represented by any rational number.
In fact, square roots forced the Pythagoreans in ancient Greece
to invent a new kind of number
which they were apparently not very happy about.
We now call these numbers, which are not equal to any rational number, irrational numbers.
The rational and irrational numbers taken together, are called the real numbers.
But more about that later.
However, even with real numbers, it is impossible to take the square root of a negative number.
This is because squaring positive or negative numbers always results in a positive number.
There is no real number which when squared, is negative.
Therefore negative numbers have no real square roots.
We saw that if we square "a" to the one-half, we get "a".
We therefore deduced that "a" to the one-half, is the square root of "a".
Likewise, if we cube "a" to the one-third
multiplying the exponents, we also get "a" to the first power
or "a".
Therefore, "a" to the one-third must be the cube root of "a".
Just like square roots, cube roots can be thought of geometrically.
If you want to construct a cube with a volume, V
you need to know what length to make the sides.
We know that the length x, cubed is equal to the volume V
so we want to find the value of x that when cubed, gives us V.
In other words, we want to find the cube root of V.
For example, if we want a volume of eight cubic meters
then a side length of two meters will work, since two cubed is eight.
Therefore, we say that two is the cube root of eight.
Cube roots, like square roots, can sometimes be integers.
For example, the cube roots of one, eight, twenty-seven, and sixty-four
are one, two, three, and four, respectively.
However, just like with square roots, the cube roots of most numbers are irrational.
However, unlike square roots
positive and negative numbers each have exactly one cube root.
For example, positive eight has the single cube root, positive two
and negative eight has the single cube root, negative two.
Positive and negative numbers have only one cube root
because when we cube a negative number we always get a negative result
and when we cube a positive number we always get positive result.
As you recall, this is not what happens when we square numbers.
Both positive and negative numbers, when squared, produce a positive result.
When we square numbers we never get a negative result.
Therefore, the square root of four can be either negative two or positive two
since either of these numbers squared produce four.
Whereas, the cube root of eight can only be positive two
and the cube root of negative eight can only be negative two.
We saw that because "a" to the one-half, squared is "a"
"a" to the one-half is the square root of "a".
And since "a" to the one-third, cubed is "a"
"a" to the one-third is the cube root of a.
Like the exponents one-half and one-third
this same logic applies to any fractional exponent, one over n.
"a" with an exponent of one over any positive integer, n
is the nth root of "a" since this is the number which, when raised to the nth power
is equal to "a".
To write the nth root of "a", we can either write "a" with an exponent one over n
or we can place "a" under a radical sign with the number n in the upper left.
This number is called the index.
When the index is odd we say that the root is an odd root
and when the index is even the root is called an even root.
Let’s say that we have an index of four
so we have "a" to the one-fourth or the fourth root of "a".
If our number "a" is sixteen then the fourth root would be two
since two to the fourth power, or four twos multiplied together, are sixteen.
However, four negative two’s multiplied together are also sixteen
so negative two is also a fourth root of sixteen.
Just as with square roots, there are two fourth roots of sixteen, one negative, and one positive.
In fact, every positive number "a" has two even roots, one negative and one positive.
And just as with square roots, negative numbers have no even roots.
Just as with cube roots, there is only one odd root for any number "a"
and this root will always have the same sign as "a".
For example, the fifth root of 32 is two
and the fifth root of negative 32 is negative two.
On the other hand, 64 has two sixth roots, negative two and two
and there is no sixth root of negative 64.
So far, we have seen that writing any number "a" with an exponent, one over n
represents the nth root of that number.
In the next lecture, we will see that exponents can not only be unit fractions.
They can also be rational numbers written as common fractions m over n.