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Surd

A surd is a radical that is not evaluated, or cannot be precisely evaluated. The radicand is often a constant, such as the square root of two: We know that the square root of 2 is 1.4142.. But why do we leave it as a radical and not convert it to the number? There are a couple of reasons:

It may cancel later

As you are working through a problem, you may get something like: The surds on the top and bottom cancel leaving

If you had evaluated the surds to roughly 1.414, they may get mixed in with other constants and you may not notice they cancel further down the work.

It may get raised to a power later

As you work a problem, you may, for example, end up squaring the surd: Since root 2 squared is obviously 2, the expression simplifies to If we had replaced root 2 with 1.414 earlier, we may not notice that squaring it results in exactly 2.

It preserves accuracy

Some radical expressions result in irrational numbers, with a result that has an infinite number of decimal places. For example, the square root of 2 is irrational. No matter how many decimal places you use in the answer, you can always add more and make it more accurate. We try to leave it as a surd, so so we are not pinning down the accuracy until later when we finally do evaluate it.

Other exponents and roots topics