Simplifying formulas » Square roots

What are the rules?


square root(a) + square root(a) = 2square root(a)

square root(a) + square root(b) = cannot be shortened

square root(a) × square root(b) = square root(a × b) = square root(ab) (with a ≥ 0 and b ≥ 0)

square root(a) gedeeld door square root(b) = square root(a/b) (with a ≥ 0 and b > 0)

All the basic calculation rules apply too, of course.
With the rule at the top, you can also simplify square roots.

Addition examples

6square root(5) + 3square root(5) = 9square root(5)

3square root(a) + 2square root(a) = 5square root(a)

7square root(a) – square root(a) + 2square root(b) = 6square root(a) + 2square root(b)

10square root(16) + 2square root(16) = 12square root(16) = 12 × 4 = 48


Multiplication examples

The intermediate step in the middle is not necessary.

square root(4) × square root(8) = square root(4 × 8) = square root(32)

3 square root(2) × 5 square root(4) = 3 × 5 × square root(2 × 4) = 15 square root(8)

4 square root(x) × –3 square root(7) = 4 × –3 × square root(x × 7) = –12 square root(7x)

Division example

(square root(60 a^3 b^3))/(square root(4ab)) = square root((60 a^3 b^2)/(4ab)) = square root(15 a^2 b) (with ab > 0)

When possible: remove the square root

Sometimes, it might happen that the square you get after simplifying is a square.
In that case you HAVE TO remove the square root.
–5 square root(5)0 × –7 square root(2) = –5 × –7 × square root(50 × 2) = 35 square root(100) = 35 × 10 = 350

When you multiply with the same square root, you can also remove it.
5 square root(3) × 6 square root(3) = 30 square root(3 × 3) = 30 square root(9) = 30 × 3 = 90
You can also write immediately this: 5 square root(3) × 6 square root(3) = 30 × 3 = 90