Dividing Surds

Dividing Surds

Introduction to Surds

• Surds are irrational numbers that, when expressed as a root, cannot resolve to a rational number. They involve square roots, cube roots or other roots which cannot be simplified to a whole number.
• For example, √2 is a surd because it is approximately equal to 1.414 and so this cannot be simplified to a rational number.

Dividing Surds

• When we divide surds, we are, in fact, dividing numbers that are under roots (mostly square roots).
• When dividing surds, we apply the rule: √a / √b = √(a/b).
• For instance, √8 / √2 simplifies as √(8/2), that equals √4, resulting in 2.

Rationalising the Denominator

• To make calculations easier and to aid in comparing and ordering, surds are often manipulated so the denominator is rational - a process called rationalising the denominator.
• This is achieved by multiplying both the numerator and the denominator by the conjugate of the denominator. For example, to rationalise the fraction 1/√3, we multiply both the numerator and denominator by √3, to get √3/3.

Examples

• Simplifying √18 / √2 by dividing the numbers under the roots, gives √9, which is equal to 3.
• Rationalise 2/√5 by multiplying both the numerator and denominator by √5, gives 2√5 / 5.

Conclusion

• Dividing surds and rationalising the denominator are essential techniques in handling surds in algebra.
• These techniques are commonly used to simplify calculations and to get the standard form of numerical expressions involving surds.