In this explainer, we will learn how to add and subtract square roots and how to use that to simplify expressions.
Before we can simplify radical expressions (i.e., expressions containing roots), we first need to consider what it means to simplify a radical expression and which expressions can actually be simplified. For example, we can note that, for any real number , the additive inverse property tells us that the number is real and that , or more succinctly, . We can use this to simplify radical expressions: .
We can combine this with the distributive property of multiplication over addition to combine radical expressions. For example, we recall that, for any real numbers , , and , we have . Therefore, if we want to evaluate , we can take out the shared factor of to get
Letβs see an example of applying this property to simplify an expression involving radicals.
Example 1: Simplifying the Addition and Subtraction of Square Roots
Simplify .
Answer
We first recall that the addition of real numbers is associative and commutative, so we can add the terms in this expression in any order. Thus,
We then recall that the additive identity property of the addition of real numbers tells us that, for any real number , we have . By setting , we can see that . Hence,
This method for the simplification of square roots allows us to simplify the roots of nonnegative integers and to use the additive inverse property of addition to simplify the addition and subtraction of radicals with the same base. There is another way of simplifying radical expressions, and we will see this with the following example.
Consider . We know that 3 has no square divisors greater that 1, so we cannot simplify either of the terms. Instead, we can recall that, for any real number , we have
In particular, if , we have
We can therefore rewrite the expression as follows:
We can then note that, for any real number , we have . Hence,
This gives us a link between adding and subtracting square roots with the same base and integer multiplication. We have the following result.
Property: Addition and Subtraction of Integer Multiples of Square Roots with the Same Base
For any integers , , and , where is nonnegative, we have
This result also holds true for any real numbers and by considering the distributive property of the multiplication of real numbers over the addition of real numbers, which states that, for any real numbers , , and , we have
Letβs now see an example of applying this property to simplify the sum of radicals with the same base.
Example 2: Simplifying the Addition of Two Square Roots with the Same Base
Simplify .
Answer
We recall that, for any integers , , and , where is nonnegative, we have
Therefore,
Hence,
A key point to note is that we cannot just add or subtract the bases of the square roots. Instead, we are simplifying the coefficients.
Consider . A very common error is to simplify this to , whereas, in reality, the expression cannot be simplified further. Similarly, a common error is to simplify as . Once again, we cannot combine these radicals since the radicands (the expressions under the root) are different.
In our next example, we will consider simplifying the sum of two radical expressions.
Example 3: Simplifying the Addition of Two Algebraic Expressions Involving Square Roots
Given that and , find the value of .
Answer
We first substitute the expressions for and to see that
We can then use the commutativity and associativity of the addition of real numbers to reorder the addition to the following:
We note that , and we recall that, for any integers , , and , where is nonnegative, we have . This means that
Hence,
A further property of square roots is illustrated by the following. We first note that is the nonnegative number, which when squared gives . We note that if , then
Thus, the square of is also . Since is also nonnegative, we have the following result.
Property: The Product Property of Square Roots
If and are nonnegative real numbers, then
Letβs use this property to rewrite by noting that
It is not clear which (if any) of or is the simpler expression. We therefore need to define what we mean by a simplified expression involving square roots.
Definition: Simplified Radical Expression
If is a nonnegative integer, we can rewrite , where is the square root of the largest the perfect square factor of and is the other non-perfect square factor of . This is called the simplified form of .
It then follows that the simplified form of an expression involving square roots involves writing every term in simplified form. This allows us to say that is the simplified form of . In fact, this gives us a method of determining the simplified form of the square root of a nonnegative integer.
We can simplify some radical expressions by noting that if and divides , say , then
This means we can simplify if it has any perfect square divisors greater than 1. This means we determine the largest the perfect square factor of to simplify .
In our next example, we will simplify the sum of two radicals where the base (or radicand) of each term is different.
Example 4: Simplifying the Addition of Two Square Roots with Different Radicands
Write in the form , where is an integer.
Answer
We first note that the base of each square root is different, so we cannot simplify this expression in its current form by addition. Instead, we note that , so we can simplify by using the fact that, for any nonnegative integers and , we have . Therefore,
We can use this to rewrite the expression given in the question as
We then recall that, for any integers , , and , where is nonnegative, we have
Hence,
Finally, we note that 2 has no perfect square divisors greater than 1, so this cannot be simplified any further.
In our next example, we will simplify the sum of three radical expressions, each with a different radicand.
Example 5: Simplifying the Addition and Subtraction of Three Square Roots with Different Radicands
Simplify .
Answer
We first note that each term has a different radicand, so we cannot directly combine the terms of this expression in its current form. Instead, letβs simplify each term separately by using the fact that, for any nonnegative integers and , we have .
We first note that , so
We then note that 3 has no perfect square divisors greater than 1, so cannot be simplified further. This means we cannot simplify this term or the second term any further.
We then find that , so
Substituting these values into the given expression yields
We can then simplify this by noting that is the additive inverse of , so that
Hence,
Therefore,
In our final example, we will use this method for the simplification of radical expressions to determine the value of an unknown in an equation.
Example 6: Finding the Missing Value in the Sum of Two Square Roots with Different Radicands
Given that , find the value of .
Answer
Since the right-hand side of the equation is of the form for some real value of , we will start by trying to write the left-hand side of the equation in this form. We can do this by simplifying each term using the fact that, for any nonnegative integers and , we have . Applying this, we see that
Substituting these values into the equation gives
We can simplify the left-hand side by factoring out to get
Hence,
Letβs finish by recapping some of the important points from this explainer.
Key Points
- For any integers , , and , where is nonnegative, we have
- If and are nonnegative real numbers, then
- If is a nonnegative integer, we can rewrite , where is the square root of the largest the perfect square factor of and is the other non-perfect square factor of . This is called the simplified form of .
- If a nonnegative integer has no perfect square divisors greater than 1, then cannot be simplified.
