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 Numbers, symbols and operators grouped together - one half of an equation, eg 2bf + 2f + 3k..
\(4 \sqrt{2} + 3 \sqrt{3}\) will not simplify because the numbers inside the The square root of a number is a number which, when multiplied by itself, gives the original number. So the square root of 25 is 5 (5 脳 5 = 25)., are not the same.
Question
Simplify the following surds, if possible:
- \(2 \sqrt{3} + 6 \sqrt{3}\)
- \(8 \sqrt{3} + 3 \sqrt{2}\)
- \(2 \sqrt{5} + 9 \sqrt{5}\)
- \(8 \sqrt{3}\)
- This will not simplify because the numbers inside the square roots are not the same.
- \(11 \sqrt{5}\)
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:
- \(\sqrt{12} - \sqrt{27}\)
- \(\sqrt{48} - \sqrt{12}\)
- \(\sqrt{(4 \times 3)} - \sqrt{(9 \times 3)} = 2\sqrt{3} - 3 \sqrt{3} = - \sqrt{3}\)
- \(\sqrt{(16 \times 3)} - \sqrt{(4 \times 3)} = 4 \sqrt{3} - 2 \sqrt{3} = 2 \sqrt{3}\)
Question
Find the exact perimeter of this shape.
\(2 \sqrt{2} + 2 \sqrt{2} + 3 \sqrt{3} + 3 \sqrt{3} = (4 \sqrt{2} + 6 \sqrt{3})~\text{cm}\)