Perform operations on radicals and rationalise the denominators

1. Addition and Subtraction of Radicals

Like radicals have the same index and radicand. Only like radicals can be added or subtracted by combining their coefficients.

Example 3: Simplify √2 + √32

First simplify √32: $\sqrt{32} = \sqrt{2 \times 2 \times 2 \times 2 \times 2} = 4\sqrt{2}$

Now add: $\sqrt{2} + 4\sqrt{2} = 5\sqrt{2}$

Example 4: Simplify 2∛81 + ∛24

$2\sqrt[3]{81} = 2\sqrt[3]{3 \times 3 \times 3 \times 3} = 2 \times 3\sqrt[3]{3} = 6\sqrt[3]{3}$ $\sqrt[3]{24} = \sqrt[3]{2 \times 2 \times 2 \times 3} = 2\sqrt[3]{3}$

Therefore: $2\sqrt[3]{81} + \sqrt[3]{24} = 6\sqrt[3]{3} + 2\sqrt[3]{3} = 8\sqrt[3]{3}$

2. Multiplication of Radicals

When multiplying radicals with the same index, multiply the radicands together:

$\sqrt{a} \times \sqrt{b} = \sqrt{ab}$

Example 5: Simplify √12(√3 + √5)

$\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$

So: $2\sqrt{3}(\sqrt{3} + \sqrt{5}) = 2\sqrt{3} \times \sqrt{3} + 2\sqrt{3} \times \sqrt{5} = 2 \times 3 + 2\sqrt{15} = 6 + 2\sqrt{15}$

Example 6: Simplify (2√3 - √2)(√3 + 3√2)

$= 2\sqrt{3} \times \sqrt{3} + 2\sqrt{3} \times 3\sqrt{2} - \sqrt{2} \times \sqrt{3} - \sqrt{2} \times 3\sqrt{2}$ $= 2 \times 3 + 6\sqrt{6} - \sqrt{6} - 3 \times 2$ $= 6 + 5\sqrt{6} - 6 = 5\sqrt{6}$

3. Division of Radicals

When dividing radicals with the same index, divide the radicands:

$\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$

Example 7: Simplify √75 ÷ √12

$\frac{\sqrt{75}}{\sqrt{12}} = \sqrt{\frac{75}{12}} = \sqrt{\frac{25}{4}} = \frac{5}{2}$

Rationalising the denominator means removing radicals from the denominator of a fraction so that the denominator becomes a rational number.

Case 1: Single Radical in Denominator

Multiply both numerator and denominator by the same radical.

Example 8: Rationalise 3/√5

$\frac{3}{\sqrt{5}} = \frac{3}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{3\sqrt{5}}{5}$

Case 2: Binomial Denominator (Sum or Difference)

Multiply by the conjugate (change the sign between the terms).

Example 9: Rationalise 1/(√5 - √3)

The conjugate of √5 - √3 is √5 + √3.

$\frac{1}{\sqrt{5} - \sqrt{3}} = \frac{1}{\sqrt{5} - \sqrt{3}} \times \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} + \sqrt{3}}$ $= \frac{\sqrt{5} + \sqrt{3}}{(\sqrt{5})^2 - (\sqrt{3})^2} = \frac{\sqrt{5} + \sqrt{3}}{5 - 3} = \frac{\sqrt{5} + \sqrt{3}}{2}$

Example 10: Rationalise √5/(√5 + √3)

$\frac{\sqrt{5}}{\sqrt{5} + \sqrt{3}} = \frac{\sqrt{5}}{\sqrt{5} + \sqrt{3}} \times \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} - \sqrt{3}}$ $= \frac{\sqrt{5}(\sqrt{5} - \sqrt{3})}{5 - 3} = \frac{5 - \sqrt{15}}{2}$

Summary of Rationalising Factors

When shopping at the market or calculating costs for building materials, you may encounter situations requiring operations with radicals. For example, if a rectangular plot of land in a village has an area of 200 square metres and one side measures √50 metres, you would need to divide radicals to find the other side (200 ÷ √50), and then rationalise the denominator to express the answer in a simpler form for practical measurement.