Lesson Explainer: Adding and Subtracting Square Roots | Nagwa Lesson Explainer: Adding and Subtracting Square Roots | Nagwa

Lesson Explainer: Adding and Subtracting Square Roots Mathematics • Second Year of Preparatory School

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 𝑥, the additive inverse property tells us that the number 𝑥 is real and that 𝑥+(𝑥)=0, or more succinctly, 𝑥𝑥=0. We can use this to simplify radical expressions: 22=0.

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 𝑎, 𝑏, and 𝑐, we have 𝑎𝑏+𝑐𝑏=(𝑎+𝑐)𝑏. Therefore, if we want to evaluate 32+22, we can take out the shared factor of 2 to get 32+22=(3+2)2=52.

Let’s see an example of applying this property to simplify an expression involving radicals.

Example 1: Simplifying the Addition and Subtraction of Square Roots

Simplify 2+6+86.

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, 2+6+86=(2+8)+66=6+66.

We then recall that the additive identity property of the addition of real numbers tells us that, for any real number 𝑥, we have 𝑥𝑥=0. By setting 𝑥=6, we can see that 66=0. Hence, 6+66=6+0=6.

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 23+3. 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 𝑥, we have 2𝑥=𝑥+𝑥.

In particular, if 𝑥=3, we have 23=3+3.

We can therefore rewrite the expression as follows: 23+3=3+3+3.

We can then note that, for any real number 𝑥, we have 3𝑥=𝑥+𝑥+𝑥. Hence, 3+3+3=33.

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 𝑎, 𝑏, and 𝑐, where 𝑐 is nonnegative, we have 𝑎𝑐+𝑏𝑐=(𝑎+𝑏)𝑐.

This result also holds true for any real numbers 𝑎 and 𝑏 by considering the distributive property of the multiplication of real numbers over the addition of real numbers, which states that, for any real numbers 𝑎, 𝑏, and 𝑐, we have 𝑎×(𝑏+𝑐)=(𝑎×𝑏)+(𝑎×𝑐).

Let’s 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 125+35.

Answer

We recall that, for any integers 𝑎, 𝑏, and 𝑐, where 𝑐 is nonnegative, we have 𝑎𝑐+𝑏𝑐=(𝑎+𝑏)𝑐.

Therefore, 125+35=(12+3)5=155.

Hence, 125+35=155.

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 5+63. A very common error is to simplify this to 113, whereas, in reality, the expression cannot be simplified further. Similarly, a common error is to simplify 2+3 as 5. 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 𝑎=673 and 𝑏=773, find the value of 𝑎+𝑏.

Answer

We first substitute the expressions for 𝑎 and 𝑏 to see that 𝑎+𝑏=673+773.

We can then use the commutativity and associativity of the addition of real numbers to reorder the addition to the following: 673+773=(6+(7))+7373.

We note that 6+(7)=67=1, and we recall that, for any integers 𝑎, 𝑏, and 𝑐, where 𝑐 is nonnegative, we have 𝑎𝑐+𝑏𝑐=(𝑎+𝑏)𝑐. This means that 7373=(77)3=143.

Hence, 𝑎+𝑏=1143.

A further property of square roots is illustrated by the following. We first note that 𝑎×𝑏 is the nonnegative number, which when squared gives 𝑎×𝑏. We note that if 𝑎,𝑏0, then 𝑎×𝑏=𝑎×𝑏×𝑎×𝑏=𝑎×𝑏=𝑎×𝑏.

Thus, the square of 𝑎×𝑏 is also 𝑎×𝑏. Since 𝑎×𝑏 is also nonnegative, we have the following result.

Property: The Product Property of Square Roots

If 𝑎 and 𝑏 are nonnegative real numbers, then 𝑎×𝑏=𝑎×𝑏.

Let’s use this property to rewrite 8 by noting that 8=2×2=2×2=22.

It is not clear which (if any) of 8 or 22 is the simpler expression. We therefore need to define what we mean by a simplified expression involving square roots.

Definition: Simplified Radical Expression

If 𝑐 is a nonnegative integer, we can rewrite 𝑐=𝑎𝑏, where 𝑎 is the square root of the largest the perfect square factor of 𝑐 and 𝑏 is the other non-perfect square factor of 𝑐. This is called the simplified form of 𝑐.

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 22 is the simplified form of 8. 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 𝑎>1 and 𝑎 divides 𝑐, say 𝑐=𝑎×𝑏, then 𝑐=𝑎×𝑏=𝑎×𝑏=𝑎𝑏.

This means we can simplify 𝑐 if it has any perfect square divisors greater than 1. This means we determine the largest the perfect square factor of 𝑐 to simplify 𝑐.

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 8+2 in the form 𝑎2, where 𝑎 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 8=2×2, so we can simplify 8 by using the fact that, for any nonnegative integers 𝑎 and 𝑏, we have 𝑎×𝑏=𝑎×𝑏. Therefore, 8=2×2=22.

We can use this to rewrite the expression given in the question as 8+2=22+2.

We then recall that, for any integers 𝑎, 𝑏, and 𝑐, where 𝑐 is nonnegative, we have 𝑎𝑐+𝑏𝑐=(𝑎+𝑏)𝑐.

Hence, 22+2=(2+1)2=32.

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 1223+427.

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, let’s simplify each term separately by using the fact that, for any nonnegative integers 𝑎 and 𝑏, we have 𝑎×𝑏=𝑎×𝑏.

We first note that 12=2×3, so 12=2×3=23.

We then note that 3 has no perfect square divisors greater than 1, so 3 cannot be simplified further. This means we cannot simplify this term or the second term any further.

We then find that 27=3×3, so 427=43×3=433=123.

Substituting these values into the given expression yields 1223+427=2323+123.

We can then simplify this by noting that 23 is the additive inverse of 23, so that 2323=0.

Hence, 2323+123=123.

Therefore, 1223+427=123.

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 98+1050=𝑥2, find the value of 𝑥.

Answer

Since the right-hand side of the equation is of the form 𝑥2 for some real value of 𝑥, 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 𝑎 and 𝑏, we have 𝑎×𝑏=𝑎×𝑏. Applying this, we see that 98=94×2=922=182,1050=1025×2=1052=502.

Substituting these values into the equation gives 182+502=𝑥2.

We can simplify the left-hand side by factoring out 2 to get (18+50)2=𝑥2682=𝑥2.

Hence, 𝑥=68.

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

Key Points

  • For any integers 𝑎, 𝑏, and 𝑐, where 𝑐 is nonnegative, we have 𝑎𝑐+𝑏𝑐=(𝑎+𝑏)𝑐.
  • If 𝑎 and 𝑏 are nonnegative real numbers, then 𝑎×𝑏=𝑎×𝑏.
  • If 𝑐 is a nonnegative integer, we can rewrite 𝑐=𝑎𝑏, where 𝑎 is the square root of the largest the perfect square factor of 𝑐 and 𝑏 is the other non-perfect square factor of 𝑐. This is called the simplified form of 𝑐.
  • If a nonnegative integer 𝑐 has no perfect square divisors greater than 1, then 𝑐 cannot be simplified.

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