How does rationalizing the denominator use the form of 1?
Rationalizing the denominator is a process used to eliminate radicals or square roots from the denominator of a fraction. It involves multiplying the numerator and denominator by a form of 1 that is carefully chosen to eliminate the square root.
To explain how the form of 1 is used to rationalize the denominator, let's consider an example:
Suppose we have the fraction 1 / √2, where √2 is the square root of 2.
When we rationalize the denominator, we want to eliminate the square root from it. One way to do this is by multiplying the fraction by an appropriate form of 1. In this case, we can multiply the fraction by √2 / √2, which is equal to 1.
The reason we choose √2 / √2 is because multiplying the numerator and denominator by √2 achieves our goal of eliminating the square root from the denominator.
Let's calculate the rationalized form:
(1 / √2) * (√2 / √2) = √2 / (√2 * √2)
= √2 / 2
As you can see, by multiplying the fraction by √2 / √2, we transformed the denominator into a rational number, which is simply 2. This is the rationalized form of the original fraction.
So, to summarize, when rationalizing the denominator, we use the form of 1 by multiplying the fraction by a carefully chosen expression that eliminates the square root or radical from the denominator, resulting in a rational number.