How to Add Square Roots

Co-authored by wikiHow Staff

Last Updated: January 31, 2023

You can perform all the usual mathematical operations on square roots, including addition, subtraction, division and multiplication. But because the radical sign over the square root represents a mathematical operation already in place, the rules for adding square roots are a little different than the rules you may be used to with integers. To add square roots, you must first understand how to simplify them.

Part 1

Part 1 of 2:

Simplifying Square Roots

1
Factor each radicand into prime numbers.^{[1]
X
Research source} An easy way to factor a number is by creating a factor tree diagram. Read Do a Factor Tree for complete instructions.
- A radicand is the number under the radical sign.
- A prime number is a number that can only be divided evenly by 1 and itself,^{[2]
  X
  Research source} for example, 2, 3, 5, 7, 11, etc.
- You do NOT need to factor any coefficients. A coefficient is a number in front of the radical sign.
- Let’s say, for example, that you want to add ${\sqrt {20}}+4{\sqrt {45}}+{\sqrt {5}}+{\sqrt {7}}.$
  To do this, you need to factor $20$ as $2\times 2\times 5$ . You also need to factor $45$ as $3\times 3\times 5$ .
- If a radicand is already a prime number, it does not need to be factored. For example, since $5$ and $7$ are already prime numbers, ${\sqrt {5}}$ and ${\sqrt {7}}$ do not need to be factored.
2
Rewrite the expression. Keep all the factors under the radical sign.
- For example, after factoring the radicands, the example expression would be ${\sqrt {2\times 2\times 5}}+4{\sqrt {3\times 3\times 5}}+{\sqrt {5}}+{\sqrt {7}}.$
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3
Circle pairs of like factors under each radical. Since you are finding a square root, by pairing up like factors, you can easily simplify the expression.
- For example, ${\sqrt {2\times 2\times 5}}$ has a pair of 2s, so draw a circle around them. $4{\sqrt {3\times 3\times 5}}$ has a pair of 3s, so draw a circle around them.
4
Factor out coefficients by identifying paired factors under each radical. The square root of any pair of factors will equal the factor, because $x\times x=x^{2}$ $x\times x=x^{{2}}$ and ${\sqrt {x^{2}}}=x$ ${\sqrt {x^{{2}}}}=x$ . Place this number in front of the radical sign. If the expression already has a coefficient, multiply the two numbers.^{[3]
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Research source}
- For example:
  ${\sqrt {2\times 2\times 5}}$
  $={\sqrt {4}}{\sqrt {5}}$
  $=2{\sqrt {5}}$
  So, ${\sqrt {20}}$ simplifies to $2{\sqrt {5}}$ .
- $4{\sqrt {3\times 3\times 5}}$
  $=4\times {\sqrt {9}}{\sqrt {5}}$
  $=(4\times 3){\sqrt {5}}$
  $=12{\sqrt {5}}$
  So, $4{\sqrt {45}}$ simplifies to $12{\sqrt {5}}$ .
5
Rewrite your problem, using the simplified terms. This will make the adding process much easier.^{[4]
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Research source}
- For example:
  ${\sqrt {20}}+4{\sqrt {45}}+{\sqrt {5}}+{\sqrt {7}}$ simplifies to
  $2{\sqrt {5}}+12{\sqrt {5}}+{\sqrt {5}}+{\sqrt {7}}$

Part 2

Part 2 of 2:

Adding Square Roots

1
Place a 1 in front of any square root that doesn’t already have a coefficient. The 1 is always understood, and so is rarely written. However, when adding, writing the 1 can help you keep track of coefficients.
- A coefficient is the number in front of the radical sign.
- For example, write ${\sqrt {5}}$ as $1{\sqrt {5}}$ .
2
Check for square roots with the same radicand. You can only add square roots that have the same radicand.^{[5]
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Research source}
- The radicand is the number underneath the radical sign.
- For example, you can add the first three terms in the expression
  $2{\sqrt {5}}+12{\sqrt {5}}+{\sqrt {5}}+{\sqrt {7}}$ , because they all have the same radicand (5).
3
Add the coefficients. Only add the coefficients for terms that have the same radicand. Do NOT add the radicands.^{[6]
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Research source}
- For example, $2{\sqrt {5}}+12{\sqrt {5}}+1{\sqrt {5}}=15{\sqrt {5}}$ .
4
Add any unlike radicands to the expression. These cannot be simplified any further, and cannot be added to any other terms. The result will be your final, simplified answer.^{[7]
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Research source}
- For example, $15{\sqrt {5}}+{\sqrt {7}}$ .

Community Q&A

Question

How do I add the square root of 5 and 212?

Donagan

Top Answerer

√5 + 212 = 2.236 + 212 = 214.236.
Question

How would I add two radicals that are prime numbers?

Donagan

Top Answerer

All you can do is evaluate each radical individually and then add them together.