To simplify expressions involving surds, we simply add or subtract coefficient of irrational factors that are alike. thus
To simplify surds, you want to express them in their simplest form typically by factoring out any perfect square factors from the radicand (the number under the square root symbol).The steps to follow includes:
- Identify the perfect square factors: Look for perfect square numbers that evenly divide into the number under the square root. For example, in √72, the perfect square factor is 36 because √36 equals 6.
- Factor out the perfect square: Rewrite the surd using the perfect square factor outside the square root. For example, √72 becomes 6√2.
- If possible, simplify further: Sometimes, you can simplify the expression even more. For instance, if you have √18, you can factor out √9 from it to get 3√2.
- Repeat if necessary: If there are still perfect square factors remaining under the square root, repeat the process until you can no longer simplify.
Example
solution
(a)
(b)
The idea is to express 12 as products of two numbers. One number results to a rational number after the split. similarly, 128 can be expressed as a product of 64 and 2 such that square-root of 64 results to a rational number.
After expressing all the terms in a simplified for , then some of them will have same irrational factor and hence they can be added to make them simpler.
(c)
The idea is to express 128 into two factors such that it is possible to find the 3rd root of one of them and get a whole number. From quick inspection, 128 =64 x 2 and third-root of 64=4.
similarly we can express 250 as a product of 125 and 2. We can be able to deduce that 125 is cube of 5 and hence cube-root of 125 is a rational whole number.
Related topics
- order of surds
- Introduction to surds
- Simplifying surds
- Multiplying surds
- Rationalizing a denominator