Lesson Explainer: Surds Mathematics

In this explainer, we will learn how to use the properties of surds to simplify expressions.

The square root of a number is the nonnegative number that when multiplied by itself returns the original number. This is called the principal square root of . For example, we can calculate that since . In fact, the square root of any perfect square is an integer; for any nonnegative integer .

If is a positive integer that is not a perfect square, then we say that is a surd. Surds are irrational numbers; they cannot be written as the quotients of integers. In particular, this tells us that the decimal expansion of a surd is infinite and nonrepeating. We sometimes refer to expressions involving roots as surds (or radicals) and the expression inside the root as the radicand.

Surds appear throughout mathematics, often as exact solutions to equations. For example, if we sketch a square with sides of length 1, then we can use the Pythagorean theorem to show that the diagonals of the square have length .

We recall that the Pythagorean theorem tells us that the square of the length of the hypotenuse, in a right triangle, is equal to the sum of the squares of the lengths of the two shorter sides. So, the length of the hypotenuse must satisfy . Since is positive, we have that .

We can use the properties and operations of real numbers to evaluate and simplify expressions involving surds. For example, we can subtract surds with the same radicand (number inside the square root):

We can also take out shared factors of surds to simplify expressions:

There are two key properties that we can use to simplify surds.

We can prove these results by squaring each expression. For example, we can note that since and are positive, and

Then, using the commutativity and associativity of multiplication, we can rewrite this as

Therefore, both and are square roots of and since they are both positive, they must be equal. A similar argument can be made to prove that for positive values of and .

We can use these properties to simplify surds whose radicands contain perfect square factors. For example, consider . We can lower the radicand of this surd by representing 12 as a product of its prime factors; namely, . Therefore,

If we substitute and into the rule , we obtain

We can calculate that , so

We cannot reduce the radicand any further since 3 contains no perfect square factors greater than 1. In general, we simplify a surd by writing it in the form , where has no perfect square factors above 1.

In practice, when simplifying surds, we try to identify if the radicand (the number inside the square root) has a factor that is a perfect square, and if it does, then we try to find the largest of these (for the sake of efficiency).

For example, with the example above, rather than writing the prime factorization of 12, we could have used that .

Let’s see an example of using these properties to simplify a given surd.

In the previous example, we identified the largest perfect square factor of the radicand to simplify the surd. However, it can be difficult to determine all of the perfect square factors. In cases like this, we can always factor the radicand into primes.

If we were to apply this method to the previous question, we would first note that 108 is even and . Next, we would note that 54 is even and . Now, we can use the fact that to see that

Thus,

We have and , so

We can evaluate and we can also note that , so we can rewrite the surd as follows:

Finally, we have that , so

It is worth noting that we could have simplified this surd by only applying this once using the fact that , so

In our next example, we will simplify the product of two surds.

In our next example, we will simplify the quotient of two surds.

In our next example, we will simplify an expression involving three terms that are surds.

In our next example, we will use this process of simplifying surds to find an unknown value in an equation.

In our next example, we will simplify an expression involving multiplication of surds.

It is worth noting that we can apply these results even when we are multiplying surds with different radicands. For example, consider . We can simplify this using the result , with and . We have

We then note that , so

In our final example, we will multiply two expressions involving surds and then simplify the resulting expression.

Although this is outside the scope of this lesson, it is worth noting that the reasoning and methodology that we have applied to square roots throughout this explainer can be generalized to other orders of roots (e.g., cube roots and roots), providing that the roots have the same order.

Let’s finish by recapping some of the important points from this explainer.