Hello.
I’m Professor Von Schmohawk and welcome to Why U.
So far we have seen that roots can be written
either as a radical expression or as an exponential expression.
In either case, when writing roots
we would like to represent their value in the simplest possible way.
We have seen that roots can sometimes be integers.
For instance, the square root of 81 is nine.
It is obviously simpler to write the integer nine, than to write the square root of 81.
However, most roots are irrational numbers.
In this case, writing the root as a radical expression
is probably simpler than writing a long sequence of digits.
In fact, we would need an infinite sequence of digits
to exactly represent an irrational root.
Even so, it is often easier to comprehend the magnitude of a numerical value
especially when it is rounded off to some reasonable number of digits.
Which form is considered simpler might depend upon its use.
A farmer who needs to build a square pen to enclose fifty square feet of space
would probably not use radical expressions to specify the dimensions.
Numbers are a much simpler way to specify dimensions.
One the other hand, an engineer or scientist might prefer the radical expression
since it is not only compact
but it represents the exact value of the root without any rounding error.
But there is another way we can represent this radical expression
which is still exact, but may be even easier to work with.
Notice that we can factor the number under the radical sign as 25 times two.
Now instead of using a radical sign, let’s write this using exponents.
We saw in the chapters on exponents
that we can distribute the exponent outside the parentheses, to each term within the parentheses.
In other words, instead of taking the square root of the entire product
we can take the square root of each term separately.
So the square root of 50 is equal to the square root of 25 times the square root of two
or five times the square root of two.
So when we factor any quantity under a square root sign
we can take the square root of each factor separately.
Then, any factors which are perfect squares can be written as the root itself
eliminating the radical sign.
Let’s try another example, simplifying the square root of 300.
In other words, let's write this expression
with the smallest possible number under the radical sign.
We can factor 300 as three times 100.
Then, each factor can be written separately under its own radical sign.
Since the square root of 100 is ten
the square root of 300 can be written as ten times the square root of three.
In this case, it was easy to see that the number under the square root sign
was the product of three times the perfect square, one-hundred.
This allowed us to take the square root of one-hundred
and bring it out from under the radical sign.
But in many cases, it is not obvious if there are factors which are perfect squares.
For instance, let's say we want to simplify the square root of 3675.
To find out if this number contains any perfect square factors
we can first factor the number into all its prime factors.
Doing this, we get three, times five, times five, times seven, times seven
or three times five squared times seven squared.
We can then bring the square roots of five squared and seven squared
out from under the radical sign
and write the expression as 35 times the square root of three.
35 times the square root of three may be easier to work with
than the square root of 3675.
However, there is another advantage to writing a radical expression in this form.
Let's say that we wanted to divide this square root by the square root of 300.
Now this quotient does not look very easy to simplify.
However, remember that the top number is 35 times the square root of three.
And, as we saw in the previous example
the square root of 300 is ten times the square root of three.
So if we like, we can rewrite this quotient using these simplified roots.
In this form, it is easy to see that the numerator and denominator have a common factor
the square root of three.
We can then eliminate this common factor, leaving us with 35 tenths or seven halves.
Sometimes, the roots of expressions involving variables can be simplified.
As a simple example, let's say the expression under the radical sign is x squared.
The square root of x squared is obviously x.
But remember that the radical sign implies the principle square root
which is always positive
and the value of the variable can be positive or negative.
Therefore, when bringing a variable out from under a square root sign
we must force the result to be positive by writing the absolute value of the variable.
This is true for square roots, fourth roots, or any even root.
However, we do not have to take the absolute value for odd roots.
This is because with odd roots
when the variable has a positive value the result should be positive
and when the variable has a negative value the result should be negative.
Now that we know how to simplify roots of variables
let's use this technique to simplify radical expression with numbers and variables.
Factoring the number under the square root sign
we can collect groups of factors which can be written as perfect squares
and bring their square roots out from under the radical sign.
Since a square root is an even root
don't forget to take the absolute value of the variables x and y.
Radical expressions can be simplified
whenever factors of the expression under the radical sign can be grouped into perfect roots
which can be brought out from the radical sign.
For example, let's say we want to simplify the cube root of 72.
Factoring 72, we see that two cubed is a factor.
So this factor can be brought out from under the cube root.
Since none of the remaining factors can be grouped into a perfect cube
they cannot be brought out
and we have two times the cube root of nine.
On the other hand, let's simplify the square root of 72.
Once again factoring 72, we see that two squared and three squared are both factors.
So these two factors can be brought out from under the square root.
and we have six times the square root of two.
So the square root of 72 can be written as six times the square root of two
and the cube root of 72 can be written as two times the cube root of nine.
In both cases, we factored 72
and then grouped the factors into products, which were perfect squares or perfect cubes
and therefore could be brought out from under the cube root or square root radical signs.
We have seen that although the roots of some numbers can be represented by integers
most roots cannot be represented by integers.
In fact, they can't even be represented by rational numbers.
In the next lecture we will see how the concept of roots
starting with the discovery of square roots
forced the world to adopt a strange new kind of number, called an irrational number.