Rationalising the Denominator: Overview
Rationalising the denominator involves eliminating any square roots or irrational numbers from the denominator of a fraction. This is achieved by multiplying both the numerator and denominator by a suitable expression.
Example 1: \( \frac{2\sqrt{100} + 5\sqrt{508}}{100} \)
1. Simplify the Square Roots:
- \( \sqrt{100} = 10 \)
- \( \sqrt{508} = \sqrt{4 \times 127} = 2\sqrt{127} \)
2. Rewrite the Expression:
- \( 2\sqrt{100} + 5\sqrt{508} = 20 + 10\sqrt{127} \)
3. New Fraction:
- \( \frac{20 + 10\sqrt{127}}{100} \)
4. Final Simplification:
- Divide both terms in the numerator by 100:
- \( \frac{20}{100} + \frac{10\sqrt{127}}{100} = \frac{1}{5} + \frac{\sqrt{127}}{10} \)
Example 2: \( 100 + \sqrt{2} + 500 + \sqrt{3390} \)
1. Combine Like Terms:
- \( 100 + 500 = 600 \)
2. Simplify the Square Roots:
- \( \sqrt{3390} = \sqrt{9 \times 378} = 3\sqrt{378} \)
- Further simplify \( \sqrt{378} = \sqrt{9 \times 42} = 3\sqrt{42} \)
- So, \( \sqrt{3390} = 9\sqrt{42} \)
3. Final Expression:
- Combine all terms: \( 600 + \sqrt{2} + 9\sqrt{42} \)
4. Final Result:
- The expression is simplified and rationalised as \( 600 + \sqrt{2} + 9\sqrt{42} \).
By following these steps, you can successfully rationalise the denominator and simplify the expression effectively.