Adding and subtracting surds

Surds with the same numbers under the roots can be added or subtracted

Example

Simplify 5\sqrt{2} - 3\sqrt{2}

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

This is similar to collecting like terms in an expression.

4 \sqrt{2} + 3 \sqrt{3} will not simplify because the numbers inside the square roots, are not the same.

Question

Simplify the following surds, if possible:

  1. 2 \sqrt{3} + 6 \sqrt{3}
  2. 8 \sqrt{3} + 3 \sqrt{2}
  3. 2 \sqrt{5} + 9 \sqrt{5}
  1. 8 \sqrt{3}
  2. This will not simplify because the numbers inside the square roots are not the same.
  3. 11 \sqrt{5}

It may be necessary to simplify one or more surds in an expression first, before adding or subtracting the surds.

Example

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

Step one:

\sqrt{12} = \sqrt{4 \times 3}

=\sqrt{4} \times \sqrt{3}

= 2\sqrt{3}

Step two:

\sqrt{27} = \sqrt{9 \times 3}

=\sqrt{9} \times \sqrt{3}

= 3\sqrt{3}

So, \sqrt{12} + \sqrt{27} is:

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

Question

Simplify:

  1. \sqrt{12} - \sqrt{27}
  2. \sqrt{48} - \sqrt{12}
  1. \sqrt{(4 \times 3)} - \sqrt{(9 \times 3)} = 2\sqrt{3} - 3 \sqrt{3} = - \sqrt{3}
  2. \sqrt{(16 \times 3)} - \sqrt{(4 \times 3)} = 4 \sqrt{3} - 2 \sqrt{3} = 2 \sqrt{3}
Question
2 sq root 2 x 3 sq root 3 rectangle

Find the exact perimeter of this shape.

2 \sqrt{2} + 2 \sqrt{2} + 3 \sqrt{3} + 3 \sqrt{3} = (4 \sqrt{2} + 6 \sqrt{3})~\text{cm}