If you were asked to find √9 you would get the accurate answer of 3
On most occasions, you are unable to get an accurate answers. √2=1.41421....There is no nice way of writing
this because √2 is an irrational number. (Cannot be written as a fraction)
It is much nicer to leave it as √2. This is called surd form.
√a × √ b = √ (a×b)
√3 × √ 3 = √ (3×3)=√9=3
√5 × √ 5 = √ (5×5)=√25=5
Generally, √a × √ a = a
√3 × √ 12 = √ (3×12)=√36= 6
√3 × √ 5 = √ (3×5)=√15, this cannot be simplified any further
You can simplify fractions, in a similar way to simplifying fractions, by looking for factors
√300 = √ (100 ×3) = √100 × √3 = 10× √3 = 10√3
√48 = √(16×3)=√16×√3=4√3
It can not always be obvious how a surd simplifies, so take it step by step
√48 = √ (4 ×12) = √ 4× √12 = 2× √12 =
2× √ (4 × 3) = 2 × √4 × √3 = 2 × 2 × √3 = 4√3.
The same as above
This is easy, however many students try to use incorrect methods to speed up the work
√ 3 + √3 does not equal √ (3+3) = √ 6
√3 + √3 is 2 lots of √3 ie √3 + √3 = 2√3
√5 + 2√5 = 3√5
√7 + 7√7 - 3√7 = 8√7-3√7=5√7
√5 + √7 cannot be simplified because the surds are different. It is like trying to add x + y
2√5 + 3√7 + 4√5 - √7 = 6√5 + 2√7. This is the final answer
√ 3 + √27 does not appear to simplify.
However, √27 = √(9×3) = √9 × √3 = 3×√3
So √ 3 + √27 = √ 3 + 3√3 = 4√3
Similarly, if faced with 2√ 3 + √75
√75 = √(25×3) = √25 × √3 = 5×√3
So, 2√ 3 + √75 = 2√ 3 + 5√3 = = 7√3