Adding and Subtracting Radicals

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Adding and subtracting radicals works like combining like terms. You can only combine like radicals, which have the same number under the radical sign. Add or subtract the numbers in front, and keep the radical the same. For example, 3\sqrt{5} + 4\sqrt{5} = 7\sqrt{5}.

Before this lesson: You should know simplifying square roots and combining like terms.

What Are Like Radicals?

Think of \sqrt{5} as a label, like a variable. You know that 3x + 4x = 7x. In the same way, 3 of something plus 4 of the same thing is 7 of it. So 3\sqrt{5} + 4\sqrt{5} = 7\sqrt{5}.

Like radicals have the same radicand, the number or expression under the radical sign. Only like radicals can be combined. The number in front of a radical is its coefficient.

RadicalsLike?Why
3\sqrt{2} and 5\sqrt{2}YesSame radicand, 2
2\sqrt{x} and 7\sqrt{x}YesSame radicand, x
\sqrt{2} and \sqrt{3}NoDifferent radicands
\sqrt{8} and \sqrt{2}Yes, after simplifying\sqrt{8} = 2\sqrt{2}
Three rows. Top: √12 + √27, labeled look unlike. An arrow labeled simplify each radical points to the middle row: 2√3 + 3√3, with both √3 parts in dashed orange boxes to show they are like radicals. An arrow labeled add the numbers in front points to the bottom row: 5√3.
√12 and √27 don’t look alike, but they simplify to 2√3 and 3√3. Then add: 5√3.

The Rule for Adding and Subtracting Radicals

\begin{aligned} a\sqrt{c} + b\sqrt{c} &= (a + b)\sqrt{c} \\ a\sqrt{c} - b\sqrt{c} &= (a - b)\sqrt{c} \end{aligned}

This is the distributive property in reverse. The radical stays the same. Only the coefficients change.

Steps for adding and subtracting radicals

  1. Simplify each radical.
  2. Find the like radicals: the ones with the same radicand.
  3. Add or subtract their coefficients. Keep the radical the same.
  4. Leave unlike radicals as separate terms.

Worked Examples

Example 1: Like radicals

Simplify 3\sqrt{5} + 4\sqrt{5} and 9\sqrt{2} - 4\sqrt{2}.

\begin{aligned} 3\sqrt{5} + 4\sqrt{5} &= (3 + 4)\sqrt{5} = 7\sqrt{5} \\ 9\sqrt{2} - 4\sqrt{2} &= (9 - 4)\sqrt{2} = 5\sqrt{2} \end{aligned}

Example 2: Some radicals don’t combine

Simplify 6\sqrt{3} + 2\sqrt{7} - 4\sqrt{3}.

Only the \sqrt{3} terms are like radicals. The \sqrt{7} term stays as it is.

6\sqrt{3} + 2\sqrt{7} - 4\sqrt{3} = 2\sqrt{3} + 2\sqrt{7}

Example 3: Simplify first

Simplify \sqrt{12} + \sqrt{27}.

These look unlike. But 12 = 4 \cdot 3 and 27 = 9 \cdot 3, so both simplify to a multiple of \sqrt{3}.

\begin{aligned} \sqrt{12} + \sqrt{27} &= 2\sqrt{3} + 3\sqrt{3} \\ &= 5\sqrt{3} \end{aligned}

Example 4: Coefficients and subtraction

Simplify 4\sqrt{18} - 2\sqrt{50}.

Simplify each radical, then multiply by the number in front.

\begin{aligned} 4\sqrt{18} - 2\sqrt{50} &= 4 \cdot 3\sqrt{2} - 2 \cdot 5\sqrt{2} \\ &= 12\sqrt{2} - 10\sqrt{2} \\ &= 2\sqrt{2} \end{aligned}

Example 5: Word problem

A rectangle is \sqrt{50} meters long and \sqrt{8} meters wide. Find its exact perimeter. Then estimate it, using \sqrt{2} \approx 1.41.

The perimeter is 2 lengths plus 2 widths. Simplify first: \sqrt{50} = 5\sqrt{2} and \sqrt{8} = 2\sqrt{2}.

\begin{aligned} P &= 2\sqrt{50} + 2\sqrt{8} \\ &= 2 \cdot 5\sqrt{2} + 2 \cdot 2\sqrt{2} \\ &= 10\sqrt{2} + 4\sqrt{2} = 14\sqrt{2} \end{aligned}

The perimeter is 14\sqrt{2} meters, or about 14 \cdot 1.41 = 19.74 meters.

Common Mistakes

MistakeExample of the errorFix
Adding the radicands\sqrt{2} + \sqrt{3} = \sqrt{5}Unlike radicals don’t combine. \sqrt{2} + \sqrt{3} is already simplified.
Changing the radicand3\sqrt{2} + 4\sqrt{2} = 7\sqrt{4}Keep the radical the same: 7\sqrt{2}.
Not simplifying first\sqrt{8} + \sqrt{2} can’t be combined\sqrt{8} = 2\sqrt{2}, so the sum is 3\sqrt{2}.
Combining unlike radicals2\sqrt{3} + 3\sqrt{2} = 5\sqrt{5}The radicands differ, so the answer stays as it is.
Forgetting a coefficient of 15\sqrt{7} - \sqrt{7} = 5\sqrt{7} means 1\sqrt{7}. So 5\sqrt{7} - 1\sqrt{7} = 4\sqrt{7}.

Radicals with Variables and Other Forms

Variables in the radicand

The same rules work with variables. Assume the variables are not negative.

\begin{aligned} 5\sqrt{x} + 2\sqrt{x} &= 7\sqrt{x} \\ \sqrt{9x} + \sqrt{4x} &= 3\sqrt{x} + 2\sqrt{x} = 5\sqrt{x} \end{aligned}

Radicals mixed with whole numbers

A whole number and a radical are unlike terms. For example, 4 + 3\sqrt{2} - 1 + \sqrt{2} = 3 + 4\sqrt{2}. Combine the numbers with the numbers and the radicals with the radicals.

ExpressionCombine?Answer
7\sqrt{6} - 2\sqrt{6}Yes5\sqrt{6}
\sqrt{5} + \sqrt{20}Yes, after simplifying3\sqrt{5}
2\sqrt{3} + 3\sqrt{5}No2\sqrt{3} + 3\sqrt{5}
6 + 2\sqrt{3}No6 + 2\sqrt{3}

Multiplying radicals uses a different rule. You will learn it in the next lesson.

Practice Problems

  1. Simplify 2\sqrt{7} + 5\sqrt{7}.
  2. Simplify 10\sqrt{3} - 4\sqrt{3}.
  3. Simplify \sqrt{6} + 8\sqrt{6} - 3\sqrt{6}.
  4. Simplify 4\sqrt{5} + 3\sqrt{2} - \sqrt{5}.
  5. Simplify \sqrt{20} + \sqrt{45}.
  6. Simplify \sqrt{75} - \sqrt{12}.
  7. Simplify 3\sqrt{8} + \sqrt{32}.
  8. Simplify 2\sqrt{27} - 5\sqrt{3} + \sqrt{48}.
  9. Simplify \sqrt{16x} + 3\sqrt{x}. Assume x \ge 0.
  10. A triangle has sides \sqrt{18}, \sqrt{32} and \sqrt{50} centimeters long. Find its exact perimeter. Then estimate it, using \sqrt{2} \approx 1.41.
Show answers
  1. 7\sqrt{7}
  2. 6\sqrt{3}
  3. (1 + 8 - 3)\sqrt{6} = 6\sqrt{6}
  4. 3\sqrt{5} + 3\sqrt{2}. The \sqrt{2} term doesn’t combine.
  5. 2\sqrt{5} + 3\sqrt{5} = 5\sqrt{5}
  6. 5\sqrt{3} - 2\sqrt{3} = 3\sqrt{3}
  7. 6\sqrt{2} + 4\sqrt{2} = 10\sqrt{2}
  8. 6\sqrt{3} - 5\sqrt{3} + 4\sqrt{3} = 5\sqrt{3}
  9. 4\sqrt{x} + 3\sqrt{x} = 7\sqrt{x}
  10. 3\sqrt{2} + 4\sqrt{2} + 5\sqrt{2} = 12\sqrt{2} centimeters. That is about 12 \cdot 1.41 = 16.92 centimeters.

FAQ

Can you add radicals with different radicands?

Not directly. First simplify each radical. If they end up with the same radicand, you can add them. If not, leave the sum as it is, like \sqrt{2} + \sqrt{3}.

What are like radicals?

Like radicals have the same radicand. For example, 4\sqrt{7} and -\sqrt{7} are like radicals, but 4\sqrt{7} and 4\sqrt{5} are not.

Do you add the numbers under the radical sign?

No. Add only the numbers in front. The radicand stays the same, so 2\sqrt{3} + 5\sqrt{3} = 7\sqrt{3}.

Why do you simplify radicals before adding?

Simplifying can show like radicals that were hidden. \sqrt{12} and \sqrt{27} look different, but they are 2\sqrt{3} and 3\sqrt{3}.

Summary

  • Like radicals have the same radicand. Only like radicals combine.
  • Use a\sqrt{c} + b\sqrt{c} = (a + b)\sqrt{c}. The radical stays the same.
  • Simplify every radical first to find hidden like radicals.
  • Never add the radicands.
  • Leave unlike radicals and whole numbers as separate terms.

Previous lesson: Simplifying Square Roots

Next lesson: Multiplying Radicals (coming soon)

Back to the Algebra 1 Index

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