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.
For any positive numbers and , we have that
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.
Example 1: Simplifying a Surd
To simplify a surd, we need to write it in the form , where has no perfect square factors greater than 1. We can do this by recalling that, for any positive numbers and , .
If we look for perfect square factors of 108, we can note that and , and we may even notice that . We can simplify the square root using any of these factorizations. But since 36 is the largest perfect square factor of 108, we will start with this value, as it will simplify our calculation:
We note that , so we have
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
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.
Example 2: Simplifying a Product of Surds
To simplify the product of two surds, we could start by simplifying each surd separately; however, in this case, neither radicand has a perfect square factor greater than 1. Instead, we can split the surds using the fact that, for any positive numbers and , we have that .
Let’s start with ; we can factor 10 into primes to note that . Thus,
Next, we can rewrite in the same way, by noting that . Therefore,
Substituting these expressions into the original product yields
We can simplify by noting that and .
We cannot simplify any further since 6 has no perfect square factors greater than 1.
In our next example, we will simplify the quotient of two surds.
Example 3: Simplifying a Quotient of Surds
To simplify the quotient of two surds, we first recall that, for any positive numbers and , we have that . Substituting and into this result gives us
We note that ; therefore, our expression simplifies to .
In our next example, we will simplify an expression involving three terms that are surds.
Example 4: Simplifying the Addition and Subtraction of Three Surds with Different Radicands
To simplify this expression, we can start by simplifying each term separately. We need to write each surd in the form , where has no perfect square factors greater than 1. We can do this by recalling that, for any positive numbers and , we have that .
Let’s start by simplifying : we factor 12 to see that . So, we have that
We cannot simplify this surd any further since 3 is a prime; it has no perfect square factors greater than 1.
The second term is ; we have already noted that we cannot simplify this surd any further.
Finally, to simplify , we factor 27 into . Thus,
Substituting these values into the given expression yields
We can take out the shared factor of to obtain which simplifies to
In our next example, we will use this process of simplifying surds to find an unknown value in an equation.
Example 5: Finding the Missing Value in the Sum of Two Square Roots with Different Radicands
Given that , find the value of .
The right-hand side of the equation is a surd written in simplified form. We can find the value of by rewriting the left-hand side in the same form and then comparing the two sides.
To simplify the left-hand side of the equation, we can simplify the two surds individually. To simplify a surd, we need to write it in the form , where has no perfect square factors greater than 1. We can do this by recalling that, for any positive numbers and , we have that .
First, we can factor 8 to see that . Thus, we can simplify as follows:
Similarly, we can factor 50 to find that . Therefore,
We can use these expressions to rewrite the left-hand side of the given equation:
This simplifies to
We now have that
In our next example, we will simplify an expression involving multiplication of surds.
Example 6: Multiplying Two Radical Expressions with the Same Base
We cannot simplify either of the factors in the given expression since 7 is a prime number; it has no perfect square factors greater than 1. Instead, we can distribute into the parentheses to rewrite the expression as follows:
We note that , so
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.
Example 7: Multiplying Two Radical Expressions and Simplifying
Express in its simplest form.
To simplify this expression, we could multiply out the parentheses; however, in this case, we have the factored form of a difference of squares: with and . Therefore, we can rewrite the expression as
We can evaluate each square separately by recalling that . Therefore,
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.
- For any positive numbers and , we have that
- We can use these properties to rewrite surds in the simplified form , where has no perfect square factors greater than 1.
- When simplifying an expression of the form , it is a good idea to look for the largest perfect square factor of , as this will simplify the calculations. If this is difficult, we can always take the approach of prime factoring as a way of determining its perfect square factors.