We will learn how to express of a simple quadratic surd. We cannot express a simple quadratic surd by the following ways:
I. A simple quadratic surd cannot be equal to the sum or difference of a rational quantity and a simple quadratic surd.
Suppose, let √p a given quadratic surd.
If possible, let us assume, √p = m + √n where m is a rational quantity and √n is a simple quadratic surd.
Now, √p = m + √n
Squaring both sides, we get,
p = m^2 + 2m√n + n
m^2 +2m√n + n = p
2m√n = p - m^2 - n
√m = (p - m^2 - n)/2m, which is a rational quantity.
From the above expression we can clearly see that the value
of a quadratic surd is equal to a rational quantity which is impossible.
Similarly, we can prove that √p ≠ m - √n
Therefore, the value of a simple quadratic surd cannot be equal to the sum or difference of a rational quantity and a simple quadratic surd.
II. A simple quadratic surd cannot be equal to the sum or difference of two simple unlike quadratic surds.
Suppose, let √p be a given simple quadratic surd. If possible, let us assume √p = √m + √n are two simple quadratic surds.
Now, √p = √m + √n
Squaring both sides we get,
p = m + 2√mn + n
√mn = (p - m - n)/2, which is a rational quantity.
From the above expression we can clearly see that the value of a quadratic surd is equal to a rational quantity, which is obviously impossible, since √m and √n are two unlike quadratic surds, hence √m ∙ √n = √mn cannot be rational.
Similarly, our assumption cannot be correct i.e. √p = √m + √n does not hold.
Similarly, we can prove that, √p ≠ √m - √n.
Therefore, the value of a simple quadratic surd cannot be equal to the sum or difference of two simple unlike quadratic surds.
11 and 12 Grade Math
From Express of a Simple Quadratic Surd to HOME PAGE
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.
Apr 18, 24 02:58 AM
Apr 18, 24 02:15 AM
Apr 18, 24 01:36 AM
Apr 18, 24 12:31 AM
Apr 17, 24 01:32 PM