An area of study in mathematics is square roots. The square root allows us to represent numbers before they are squared.

Square root calculations include addition and subtraction. For multiplication and division, we can just multiply the numbers in the root symbol as they are. For addition and subtraction, on the other hand, the rule is that the numbers in the radical symbol must be the same, or the calculation is not allowed.

Addition and subtraction also require prime factorization and rationalizing the denominator. In square root calculation, the content of addition and subtraction is more difficult than multiplication.

So we will explain how to add and subtract square roots.

How to Add and Subtract Square Root

In square root multiplication and division, we can multiply the square roots by each other. Just multiply the numbers in the root symbol, although there are caveats.

On the other hand, how should we think about addition and subtraction? When it comes to addition and subtraction, you should not add or subtract between radical sign with different numbers. For example, the following are wrong.

The method of calculation is different from multiplication and division. Why can’t we add between square roots with different numbers? Because they are numbers before they are squared, so they are different in nature than integers.

For example, $\sqrt{9}=\sqrt{3^2}=3$. And $\sqrt{16}=\sqrt{4^2}=4$. In this case, we have the following.

  • $\sqrt{9}+\sqrt{16}=3+4=7$

On the other hand, what happens if we add the numbers in the radical symbols together without making them integers? The result is as follows.

  • $\sqrt{9}+\sqrt{16}=\textcolor{red}{\sqrt{9+16}}=\sqrt{25}$

52 is 25. Therefore, $\sqrt{25}=\sqrt{5^2}=5$. However, the actual answer must be 7. That means that $\sqrt{9}+\sqrt{16}=\sqrt{25}=5$ is wrong.

Since the number squared is the root, we get the following.

  • $\sqrt{36}=6$
  • $\sqrt{100}=10$
  • $\sqrt{2500}=50$
  • $\sqrt{10000}=100$

Thus, compared to integers, the numbers in the root symbol can be quite large. Think of integers (natural numbers) and numbers in the radical symbol are completely different. This is why adding or subtracting directly to the numbers in the root sign is a mistake.

Addition and subtraction Can Be Done When the Numbers in a Radical Symbol Are Same

How can we add and subtract with square roots? The way it works is that we can only add and subtract if the numbers in the root symbol are the same. For example, it is the following.

  • $4\sqrt{2}-\sqrt{2}=3\sqrt{2}$

In this case, the numbers in the radical sign are common to 2. Therefore, it is possible to add and subtract the integers before the root sign.

$4\sqrt{2}$ means that there are four $\sqrt{2}$. So if we subtract one $\sqrt{2}$, we have three $\sqrt{2}$ left.

This is why if the numbers in the root sign are the same, we can add and subtract.

On the other hand, what happens if the numbers in the root symbol are different? As we already explained, if the numbers in the radical symbol are the same, we can add and subtract. However, if the numbers in the radical symbol are different, we cannot add or subtract them. For example, the following calculation is done.

  • $2\sqrt{\textcolor{red}{2}}-\sqrt{3}+3\sqrt{\textcolor{red}{2}}=5\sqrt{2}-\sqrt{3}$

We cannot add or subtract numbers with different properties. Therefore, the answer to this calculation is $5\sqrt{2}-\sqrt{3}$. To distinguish if addition and subtraction are possible, see if the numbers in the root symbol are the same or not.

-The Calculation Method is the same as the Algebraic Expression

The calculation of a square route is the same way as in the algebraic expression. In algebraic expressions, even different alphabets can be multiplied and divided. However, different letters cannot be added or subtracted, as shown below.

  • $2x+x-2y+4y=3x+2y$

Square roots can also be multiplied and divided, even if the numbers in the root symbol are different. However, if the numbers in the radical sign are different, we cannot add or subtract, as shown below.

  • $2\sqrt{\textcolor{red}{2}}+\sqrt{\textcolor{red}{2}}-2\sqrt{\textcolor{blue}{3}}+4\sqrt{\textcolor{blue}{3}}=3\sqrt{2}+2\sqrt{3}$

Even if the properties are different, we can still do multiplication and division. But if they are different in properties, we cannot add or subtract. In mathematics, make sure to understand this rule.

Match the Numbers in the Root Symbol by Prime Factorization

Once you understand the rules we’ve discussed, you will be able to add and subtract square roots. However, before adding and subtracting, in many cases, we must do prime factorization beforehand. Prime factorization will make the numbers in the radical sign clearer.

When calculating the square root, we must do prime factorization to form $a\sqrt{b}$. For example, we have the following.

$\sqrt{18}+\sqrt{50}-\sqrt{32}$

$=\sqrt{3^2×2}+\sqrt{5^2×2}-\sqrt{4^2×2}$

$=3\sqrt{2}+5\sqrt{2}-4\sqrt{2}$

$=4\sqrt{2}$

Prime factorization is important in the calculation of square roots because it allows us to minimize the number in the radical sign. As a result, addition and subtraction are available.

After Rationalizing the Denominator, Making the Common Denominator and Calculate It

There is another important procedure when doing the square route calculation. It is rationalizing the denominator. If the denominator has square roots (irrational numbers), it cannot be calculated. So, by rationalizing the denominator, if we change the number of denominators to integers, we can add and subtract square roots from each other by creating a common denominator.

For example, how can we do the following calculation?

  • $\displaystyle\frac{1}{\sqrt{3}}+\sqrt{3}$

In mathematics, the answer is incorrect if the denominator has a root. The reason for this is that rationalizing the denominator allows us to make the numbers simpler.

In the case of rationalizing the denominator, we can calculate the following.

$\displaystyle\frac{1}{\sqrt{3}}+\sqrt{3}$

$=\displaystyle\frac{1\textcolor{red}{×\sqrt{3}}}{\sqrt{3}\textcolor{red}{×\sqrt{3}}}+\sqrt{3}$

$=\displaystyle\frac{\sqrt{3}}{3}+\sqrt{3}$

$=\displaystyle\frac{\sqrt{3}}{3}+\textcolor{red}{\displaystyle\frac{3\sqrt{3}}{3}}$ (Making a common denominator)

$=\displaystyle\frac{4\sqrt{3}}{3}$

Rationalizing the denominator in this way allows for addition and subtraction by the common denominator.

Square Root Multiplication and Division Using the Distributive Property

When we actually do square route calculations, we don’t just add and subtract. In many cases, the equation includes multiplication and division. Therefore, we should be able to calculate expressions that include a mixture of addition, subtraction, multiplication, and division.

As mentioned earlier, algebraic expressions and square roots are calculated the same way. An important rule in algebraic expressions is the distributive property. Here is the distributive property.

The distributive property is a method for removing the parentheses in multiplication. We should use the distributive property to calculate the square root as well.

For example, how can we calculate the following.

  • $2(1+\sqrt{2})$

To solve this problem, let’s make use of the distributive property. It looks like the following.

After expanding the equation, proceed with the calculation by adding and subtracting.

Expanding the Equation with the Factoring Formulas

However, in the calculation of the square root, we have to calculate a more complex equation. We use the factoring formulas to expand the formula.

When expanding a multiplication equation with two parentheses, we use the following formula.

Other factoring formulas are also used when calculating more efficiently. The following four formulas must be remembered in mathematical calculations.

  • $(x+a)(x+b)=x^2+(a+b)x+ab$
  • $(x+a)^2=x^2+2ax+a^2$
  • $(x-a)^2=x^2-2ax+a^2$
  • $(x+a)(x-a)=x^2-a^2$

By using these formulas, we can calculate the square root. For example, let’s consider how we can do the following calculation.

  • $(3+\sqrt{5})^2$

In order to do this calculation, we must use the factoring formula. The calculation is as follows.

$(3+\sqrt{5})^2$

$=3^2+2×3×\sqrt{5}+(\sqrt{5})^2$

$=9+6\sqrt{5}+5$

$=14+6\sqrt{5}$

Integers and routes are different in properties. Therefore, 14 and $6\sqrt{5}$ cannot be calculated further. So, the calculation is complete. Let’s try to understand the nature of the square root so that we can calculate it.

Exercises: Addition and Subtraction of Square Route Calculation

Q1: Do the following calculations.

  1. $\sqrt{75}+\sqrt{98}-\sqrt{108}$
  2. $\sqrt{8}-\displaystyle\frac{1}{\sqrt{2}}$
  3. $\sqrt{6}(\sqrt{2}-\sqrt{3})+3\sqrt{3}$

A1: Answers.

(a)

After doing prime factorization, let’s do the square root calculation.

$\sqrt{75}+\sqrt{98}-\sqrt{108}$

$=5\sqrt{3}+7\sqrt{2}-6\sqrt{3}$

$=7\sqrt{2}-\sqrt{3}$

(b)

After rationalizing the denominator, let’s do the square route calculation.

$\sqrt{8}-\displaystyle\frac{1}{\sqrt{2}}$

$=2\sqrt{2}-\displaystyle\frac{1\textcolor{red}{×\sqrt{2}}}{\sqrt{2}\textcolor{red}{×\sqrt{2}}}$

$=2\sqrt{2}-\displaystyle\frac{\sqrt{2}}{2}$

$=\displaystyle\frac{4\sqrt{2}}{2}-\displaystyle\frac{\sqrt{2}}{2}$

$=\displaystyle\frac{3\sqrt{2}}{2}$

(c)

Let’s calculate using the distributive property.

$\sqrt{6}(\sqrt{2}-\sqrt{3})+3\sqrt{3}$

$=\sqrt{12}-\sqrt{18}+3\sqrt{3}$

$=2\sqrt{3}-3\sqrt{2}+3\sqrt{3}$

$=-3\sqrt{2}+5\sqrt{3}$


Q2: Do the following calculation.

For $x=2+\sqrt{2}$ and $y=4+2\sqrt{2}$, calculate the value of $xy-2x^2$.

A2: Answer.

For $x$ and $y$, we can calculate by substituting numbers. The result is as follows.

  • $(2+\sqrt{2})(4+2\sqrt{2})-2(2+\sqrt{2})^2$

But the math is hard. Isn’t there an easier way to calculate it? So, let’s use factorization. Factoring $xy-2x^2$ gives us the following.

  • $xy-2x^2=x(y-2x)$

Then substitute into this equation. It is as follows.

$x(y-2x)$

$=(2+\sqrt{2})\{4+2\sqrt{2}-2(2+\sqrt{2})\}$

$=(2+\sqrt{2})(4+2\sqrt{2}-4-2\sqrt{2})$

$=(2+\sqrt{2})×0$

$=0$

Adding and Subtracting Square Root in Mathematics

When we learn the basics of square roots, after learning multiplication and division, we learn addition and subtraction. It’s not just a matter of adding the numbers in the root symbol; there are rules to root calculation.

Adding and subtracting square root is the same as calculating an algebraic expression. For multiplication and division, we can multiply as is. On the other hand, if we want to find the answer to addition and subtraction, the numbers in the radical sign must be the same. The properties must be the same or we cannot do addition and subtraction.

Therefore, we need to do prime factorization and rationalizing the denominator so that the numbers in the root symbol are aligned. Also, try to use factoring formulas for multiplication equations with parentheses.

When learning square roots, one of the most difficult concepts to learn is addition and subtraction. Make sure you understand why we cannot add and subtract unless the numbers in the root symbol are the same before calculating the square root.