Adding and subtracting surds

Adding and subtracting surds

There are 3 apples in a basket. We add 5 apples, and now we have 8 apples. But if we add 5 pears instead, we don't get 8 apples or 8 pears. We still have 3 apples and 5 pears.

Surds work the same way. 3√2 + 5√2 = 8√2, because we are adding the same "things": roots of two. But √2 + √3 cannot be written as a single term. (In the US, surds are called radicals, and the same rules apply.)

Contents

Like surds

Terms with the same root are called like surds. We combine them by adding or subtracting the numbers in front of them. The root stays the same.

a√r + b√r = (a + b)√r

On its own, √2 means 1 · √2. We just don't write the 1.

Worked example 1

3√2 + 5√2 − √2

  1. Every term has √2.
  2. The numbers in front of the root: 3 + 5 − 1 = 7.
  3. 3√2 + 5√2 − √2 = 7√2.

Simplify first

Surds that look different can turn out to be the same after simplifying.

Worked example 2

√12 + √27

  1. √12 = √(4 · 3) = 2√3.
  2. √27 = √(9 · 3) = 3√3.
  3. 2√3 + 3√3 = 5√3.

Worked example 3

√50 − √18

  1. √50 = √(25 · 2) = 5√2.
  2. √18 = √(9 · 2) = 3√2.
  3. 5√2 − 3√2 = 2√2.

More terms and numbers in front of the root

If there is a number in front of the root, multiply it by what you take out of the root.

Worked example 4

2√48 − 3√12 + √3

  1. 2√48 = 2 · √(16 · 3) = 2 · 4√3 = 8√3.
  2. 3√12 = 3 · √(4 · 3) = 3 · 2√3 = 6√3.
  3. 8√3 − 6√3 + √3 = (8 − 6 + 1)√3 = 3√3.

Worked example 5

√8 + √12 − √18

  1. √8 = 2√2, √12 = 2√3, √18 = 3√2.
  2. 2√2 + 2√3 − 3√2 = −√2 + 2√3.
  3. √2 and √3 are different roots, so we cannot combine them any further. The answer: 2√3 − √2.

Cube roots

The method is the same. When simplifying, you just look for cubes (8, 27, 64, 125…).

Worked example 6

∛16 + ∛54

  1. ∛16 = ∛(8 · 2) = 2∛2.
  2. ∛54 = ∛(27 · 2) = 3∛2.
  3. 2∛2 + 3∛2 = 5∛2.

⚠️ You cannot combine a square root with a cube root: √2 + ∛2 stays as it is.

Common mistakes

  • Adding the numbers under the root. √2 + √3 is not √5. Check: √2 + √3 ≈ 1.41 + 1.73 = 3.14, but √5 ≈ 2.24.
  • Forgetting to simplify. √12 + √27 looks like a sum you cannot combine, but it equals 5√3.
  • Forgetting the hidden 1. 5√3 − √3 = 4√3, not 5.
  • Forgetting the number in front of the root. 2√48 = 2 · 4√3 = 8√3, not 4√3.

Summary

  • You can only combine like surds (the same index of the root and the same radicand).
  • a√r + b√r = (a + b)√r. You add the numbers in front of the root.
  • Simplify every surd before adding.
  • Leave terms with different roots side by side.

Try it yourself

  1. 4√5 − 7√5 + 2√5
  2. √20 + √45
  3. 3√8 − √32
  4. 2∛24 − ∛81
Answers:
  1. (4 − 7 + 2)√5 = −√5.
  2. 2√5 + 3√5 = 5√5.
  3. 3 · 2√2 − 4√2 = 6√2 − 4√2 = 2√2.
  4. 2 · 2∛3 − 3∛3 = 4∛3 − 3∛3 = ∛3.

Practice

👉 Next: Multiplying and dividing surds