Surds

Simplifying Surds

A surd is a square root of a number that cannot be reduced to a whole number. For example, √4 isn't a surd as it can be reduced to 2, but √5 is a surd as it cannot be reduced to a whole number. However, some surds can be simplified. The simplest example of this is √8, as it is equivalent to √(4*2), which equals (√2)*(√4), which then equals 2√2. Thus, √8 can be simplified to 2√2.

Adding or Subtracting Surds

This can be used to add up surds. For example, √12 + 2√27 can be expressed as (√4)*(√3) + 2((√9)(√3)), which equals 2√3 + 6√3, giving the overall answer of 8√3. The key point is that the two surds both share a common surd, and so they can be added or subtracted from one another.

Rationalising the Denominator

Say you have the fraction ¹⁄_(√2+3). At first glance, it doesn't look like that it can be simplified, but it can!. By multiplying both the numerator and the denominator by ^(√2-3)⁄_(√2-3) (note: √2-3 is called the conjugate of √2+3, basically the sign is flipped). It doesn't affect its value but it now can be expressed as ^√2-3⁄_{(√2+3)*(√2-3)}. Expanding the bracket on the denominator gives the new fraction ^√2-3⁄_(2-9) = ^√2-3⁄_-7 = ^3-√2⁄₇ = ¹⁄₇ (3-√2).

This can also be done with more complicated fractions, for example ^(2√3+√2)⁄_(3√2+√3) = ^(2√3+√2)⁄_(3√2+√3) * ^(3√2-√3)⁄_(3√2-√3) = ^5√6⁄₁₅ = ^√6⁄₃.

Summary

• Surds are a square root of a number that cannot be reduced to a whole number.
• Surds can be simplified if the number inside it is a multiple of a square number.
• Some surds can be added or subtracted if when simplified the number inside the square root is the same.
• A fraction with a surd on the bottom can be simplifed by multiplying both the numerator and denominator by its conjugate and expanding the brackets.