Calculating Exactly with Fractions Surds and Multiples of Pi

Exact Calculation with Fractions

When working on the TMUA, it is standard practice to leave answers in fractional form unless a decimal or percentage is specifically requested. To calculate exactly with fractions, you must be proficient in finding the lowest common multiple (LCM) of denominators to perform addition and subtraction.

Worked Example: Combining Multiple Fractions

Calculate exactly: $\frac{2}{3} + \frac{3}{5} - \frac{7}{12}$

1. Find a common denominator. The LCM of 3, 5, and 12 is 60. Alternatively, you could use $3 \times 5 \times 12 = 180$, though this requires more simplification later.
2. Convert each fraction: $\frac{2}{3} = \frac{40}{60}$, $\frac{3}{5} = \frac{36}{60}$, and $\frac{7}{12} = \frac{35}{60}$.
3. Combine the numerators: $\frac{40 + 36 - 35}{60} = \frac{41}{60}$.

Calculating with Surds

A surd is an expression involving roots (usually square roots) that cannot be simplified to a rational number (an integer or fraction). Examples include $\sqrt{2}$, $1 + 2\sqrt{3}$, and $3\sqrt{7}$.

Fundamental Rules for Surds

Surds follow specific algebraic rules for multiplication and division, but they cannot be combined simply through addition or subtraction:

• Multiplication: $\sqrt{x}\sqrt{y} = \sqrt{xy}$
• Division: $\sqrt{x} \div \sqrt{y} = \sqrt{\frac{x}{y}}$
• Addition: $\sqrt{x} + \sqrt{y} \neq \sqrt{x + y}$ (This is a common misconception).
• Subtraction: $\sqrt{x} - \sqrt{y} \neq \sqrt{x - y}$.

Simplifying Surd Expressions

Surds can often be simplified by identifying the largest square factor within the radicand (the number under the root). For example: $\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4}\sqrt{3} = 2\sqrt{3}$.

Worked Example: Simplifying and Multiplying

Calculate exactly: $3\sqrt{3} \times 7\sqrt{12}$

1. Simplify the second surd: $7\sqrt{12} = 7\sqrt{4 \times 3} = 7 \times 2\sqrt{3} = 14\sqrt{3}$.
2. Multiply the expressions: $3\sqrt{3} \times 14\sqrt{3} = (3 \times 14) \times (\sqrt{3} \times \sqrt{3})$.
3. Evaluate: $42 \times 3 = 126$.

Another example: $\sqrt{21} \times \sqrt{28} = \sqrt{3 \times 7} \times \sqrt{4 \times 7} = \sqrt{3}\sqrt{7} \times 2\sqrt{7} = 2 \times 7 \times \sqrt{3} = 14\sqrt{3}$.

Rationalising the Denominator

Rationalising the denominator means removing any surds from the bottom of a fraction so that the denominator is a rational number. The method depends on the form of the denominator.

Type 1: Single Surd Denominator

If the denominator is $\sqrt{a}$, multiply both the numerator and denominator by $\sqrt{a}$. Example: $\frac{3}{\sqrt{7}} = \frac{3 \times \sqrt{7}}{\sqrt{7} \times \sqrt{7}} = \frac{3\sqrt{7}}{7}$.

Type 2: Binomial Denominator ($x + \sqrt{y}$ or $x - \sqrt{y}$)

To rationalise these, multiply by the conjugate (the same expression but with the opposite sign). This creates a difference of two squares: $(a + b)(a - b) = a^2 - b^2$.

Example: $x + \sqrt{y}$ form Rationalise $\frac{5}{3 + 2\sqrt{5}}$:

1. Multiply by $\frac{3 - 2\sqrt{5}}{3 - 2\sqrt{5}}$.
2. Denominator: $(3 + 2\sqrt{5})(3 - 2\sqrt{5}) = 3^2 - (2\sqrt{5})^2 = 9 - (4 \times 5) = 9 - 20 = -11$.
3. Result: $\frac{5(3 - 2\sqrt{5})}{-11} = \frac{15 - 10\sqrt{5}}{-11} = \frac{10\sqrt{5} - 15}{11}$.

Example: $\sqrt{x} - \sqrt{y}$ form Rationalise $\frac{3}{\sqrt{5} - \sqrt{2}}$:

1. Multiply by $\frac{\sqrt{5} + \sqrt{2}}{\sqrt{5} + \sqrt{2}}$.
2. Denominator: $(\sqrt{5} - \sqrt{2})(\sqrt{5} + \sqrt{2}) = 5 - 2 = 3$.
3. Result: $\frac{3(\sqrt{5} + \sqrt{2})}{3} = \sqrt{5} + \sqrt{2}$.

Type 3: Complex Multi-step Rationalisation

Rationalise $\frac{\sqrt{5}}{\sqrt{15}(\sqrt{7} - \sqrt{2})}$:

1. Address the square root outside the bracket first: $\frac{\sqrt{5}}{\sqrt{15}(\sqrt{7} - \sqrt{2})} \times \frac{\sqrt{15}(\sqrt{7} + \sqrt{2})}{\sqrt{15}(\sqrt{7} + \sqrt{2})}$.
2. Simplify denominator: $15(7 - 2) = 15 \times 5 = 75$.
3. Simplify numerator: $\sqrt{5}\sqrt{15}(\sqrt{7} + \sqrt{2}) = \sqrt{75}(\sqrt{7} + \sqrt{2}) = 5\sqrt{3}(\sqrt{7} + \sqrt{2})$.
4. Final fraction: $\frac{5\sqrt{3}(\sqrt{7} + \sqrt{2})}{75} = \frac{\sqrt{3}(\sqrt{7} + \sqrt{2})}{15}$.

Multiples of $\pi$

Calculating exactly with $\pi$ means leaving the symbol in your final answer rather than substituting $3.14$ or $\frac{22}{7}$.

Worked Example: Circle Area A circle has a radius of 4 cm. Find its exact area. Using Area $= \pi r^2$: Area $= \pi \times 4^2 = 16\pi$ cm². This is the exact value required for the exam.