How do you calculate surds? And what does it have to do with Pythagoras theorem?

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embizze | High School Teacher | (Level 2) Educator Emeritus

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A surd is basically a radical expression that cannot be written as a rational number. Thus `sqrt(2)` is a surd while `sqrt(4)=2` is not a surd.

Typically in mathematics we write surds in simplified form. A radical is simplified if:

(1) There are no fractions in the radicand.

(2) There are no radicals in the denominator of a fraction.

(3) There are no perfect `n^(th)` powers in the radicand where n is the index. (If there is no index, it is understood to be 2. Thus `sqrt(2)` has index 2, while the index of `root(3)(2)` is 3.)

You will encounter surds when using the Pythagorean theorem because the theorem deals with the squares of the lengths of the sides. To compute the side length, you must take the square root.

For example, if the legs of a right triangle are 4 and 5, then the hypotenuse c can be found:

`4^2+5^2=c^2 ==> c^2=41 ==> c=sqrt(41)`

Sometimes the radical simplifies: Suppose the legs are 5 and 5. Then:

`5^2+5^2=c^2 ==> c^2=50 ==>c=sqrt(50) ==> c=5sqrt(2)`