Rationalising the Denominator

Rationalising The Denominator

• Rationalising the denominator means eliminating the surds (radicals) from the denominator of a fraction.
• This is done to simplify the fraction and make it easier to work with.
• The process involves multiplying the numerator and denominator by the conjugate of the denominator.
• The conjugate of a surd is a value that, when multiplied by the surd, eliminates the surd.

Example: Rationalise the denominator of 1/(√2 + 1):

• Multiply numerator and denominator by √2 - 1 (conjugate of √2 + 1)
• 1/(√2 + 1) = (√2 - 1)/(√2 + 1)(√2 - 1) = (√2 - 1)/((√2)^2 - 1^2) = (√2 - 1)/1

Simplifying Surds

• Simplifying surds involves expressing them in their simplest form.
• This is done by removing any perfect squares from the surd.
• A perfect square is a value that can be expressed as n^2, where n is an integer.

Example: Simplify √8:

• √8 = √(4 × 2) = √4 × √2 = 2√2

Addition And Subtraction Of Surds

• Surds can be added and subtracted only if they have the same value under the surd.
• When adding or subtracting surds, combine like terms.

Example: Simplify 2√3 + 3√3:

• 2√3 + 3√3 = (2 + 3)√3 = 5√3

Multiplication Of Surds

• Surds can be multiplied by multiplying the values under the surd and adding the indices.
• When multiplying surds, the order of the surds does not matter.

Example: Simplify √2 × √3:

• √2 × √3 = √(2 × 3) = √6

Rationalising Cube Roots

• Rationalising cube roots involves eliminating the cube roots from the denominator of a fraction.
• This is done by multiplying the numerator and denominator by the conjugate of the denominator.
• The conjugate of a cube root is a value that, when multiplied by the cube root, eliminates the cube root.

Example: Rationalise the denominator of 1/(∛2 + 1):

• Multiply numerator and denominator by ∛4 - 1 (conjugate of ∛2 + 1)
• 1/(∛2 + 1) = (∛4 - 1)/(∛2 + 1)(∛4 - 1) = (∛4 - 1)/((∛2)^3 + 1^3) = (∛4 - 1)/3

Rationalising the Denominator

• Rationalising the denominator is the process of removing surds from the denominator of a fraction to simplify it and make it easier to work with.
• The goal is to get a rational denominator, i.e., a denominator without surds.

Steps to Rationalise the Denominator

• Multiply the numerator and denominator by the conjugate of the denominator.
• Simplify the resulting fraction.

Simplifying Surds

• Surds are irrational numbers that cannot be expressed as a finite decimal or fraction.
• Simplifying surds involves expressing them in their simplest form by: • Removing any perfect squares from the surd. • Combining like terms.

Operations with Surds

• Surds can be added or subtracted only if they have the same order (i.e., same radical index).
• The rules for adding and subtracting surds are similar to those for adding and subtracting algebraic terms.
• Example: √2 + √2 = 2√2, but √2 + √3 cannot be simplified further.

Multiplication of Surds

• Surds can be multiplied by multiplying the numbers inside the surd and applying the rules of indices.
• Example: √2 × √3 = √(2 × 3) = √6
• When multiplying surds with different orders, the resulting surd will have the highest order.

Rationalising Cube Roots

• Rationalising cube roots involves removing cube roots from the denominator of a fraction.
• This can be done by multiplying the numerator and denominator by the conjugate of the denominator, which is the cube of the denominator.
• Example: 1 / (√2) = (√2^2) / (√2^3) = (√4) / 2