Express of a Simple Quadratic Surd

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

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