About rationalizing denominator, is it true that you can't always multiply by the same square root denominator to get a perfect square?
yes, because (√n)² = n
so, multiplying the root by itself gets rid of the root.
If the denominator is not just a root, but something like √m + n, then you need to multiply by √m - n, since
(√m + n)(√m - n) = m - n²
also, you sometimes have to use
(√m + √n)(√m - √n) = m - n
That's correct! In some cases, multiplying both the numerator and denominator by the same square root can help rationalize the denominator and make it a perfect square. However, there are certain situations where this method might not work.
To explain further, let's take an example. Consider the fraction 1/sqrt(2). If we multiply both the numerator and denominator by sqrt(2), we get sqrt(2)/2, which is a perfect square. So, in this case, we can multiply by the square root of the denominator to rationalize it.
However, let's consider another example: 1/sqrt(3). If we multiply both the numerator and denominator by sqrt(3), we end up with sqrt(3)/3. In this case, the denominator is not a perfect square, and multiplying by the square root of the denominator does not rationalize it.
In situations like this, where the denominator is not a perfect square, a different method called "rationalizing the denominator by conjugate" can be used. This involves multiplying both the numerator and denominator by the conjugate of the denominator. For the fraction 1/sqrt(3), we can multiply both the numerator and denominator by sqrt(3) to get sqrt(3)/3, which is now a rationalized form.
So, while multiplying by the square root of the denominator can work for some cases, it is not always the best approach. The conjugate method provides a more reliable way to rationalize the denominator when the denominator is not a perfect square.