Exact Calculations with Fractions Surds and Pi

Working with Fractions

When calculating exactly with fractions, the standard practice is to leave the final result as a fraction. Unless a question specifically requests a conversion to decimals or percentages, you should maintain the fractional form to preserve accuracy.

Worked Example: Calculating with Fractions

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

To solve this, we must place all fractions over a common denominator. The Lowest Common Multiple (LCM) of 3, 5, and 12 is 60. Alternatively, you could use the product $3 \times 5 \times 12 = 180$, though 60 is more efficient.

$\frac{2}{3} + \frac{3}{5} - \frac{7}{12} = \frac{40}{60} + \frac{36}{60} - \frac{35}{60}$

Adding and subtracting the numerators gives:

$\frac{40 + 36 - 35}{60} = \frac{41}{60}$

Calculating with Surds

A surd is an expression containing square roots, cube roots, or higher roots that cannot be simplified into an integer or a fraction. Examples of surds include $\sqrt{2}$, $1 + 2\sqrt{3}$, and $3\sqrt{7}$.

There are specific rules for combining surds:

1. Multiplication and Division: Surds can be combined under a single root. For example, $\sqrt{x}\sqrt{y} = \sqrt{xy}$ and $\sqrt{x} \div \sqrt{y} = \sqrt{\frac{x}{y}}$.
2. Addition and Subtraction: Surds cannot be combined in the same way. It is a common error to assume $\sqrt{x} + \sqrt{y} = \sqrt{x + y}$ or $\sqrt{x} - \sqrt{y} = \sqrt{x - y}$: these operations are not valid.

Worked Example: Calculating with Surds

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

First, we can simplify $\sqrt{12}$ into $\sqrt{4}\sqrt{3}$:

$3\sqrt{3} \times 7\sqrt{3}\sqrt{4} = 3\sqrt{3} \times 7\sqrt{3} \times 2$

Rearranging the terms: $3 \times 7 \times 2 \times (\sqrt{3} \times \sqrt{3}) = 42(\sqrt{3})^2$

Since $(\sqrt{3})^2 = 3$, the calculation becomes: $42 \times 3 = 126$

Multiples of Pi

Calculating exactly with multiples of $\pi$ requires leaving the answer in terms of the symbol $\pi$. You must not substitute numerical approximations such as $\pi \approx 3.14$ or $\pi \approx 3$.

Worked Example: Area of a Circle

The radius of a circle is 4 cm. What is the exact value of the area of the circle in cm$^2$?

Using the formula for the area of a circle, $A = \pi r^2$:

$A = \pi \times 4^2$ cm$^2 = 16\pi$ cm$^2$

This result is the exact value and should not be converted to a decimal.

Simplifying Surds

Surds are simplified by identifying the largest square factor within the radicand (the number under the root). We then split the root and simplify the square root of the square factor.

Worked Examples: Simplifying Surds

1. Simplify $\sqrt{128}$. Since 64 is a square factor of 128: $\sqrt{128} = \sqrt{64}\sqrt{2} = 8\sqrt{2}$

2. Simplify $\sqrt{21} \times \sqrt{28}$. Break each surd into its prime factors or identifying square factors: $\sqrt{3 \times 7} \times \sqrt{4 \times 7} = \sqrt{3}\sqrt{7}\sqrt{4}\sqrt{7}$

Reorganising the terms: $\sqrt{3} \times 2 \times (\sqrt{7})^2 = \sqrt{3} \times 2 \times 7 = 14\sqrt{3}$

Rationalising the Denominator

To rationalise the denominator means to rewrite a fraction so that it no longer contains a surd in the denominator. The method used depends on the structure of the denominator.

Type 1: Single Surd Denominator

If the denominator is a single surd, multiply both the numerator and the denominator by that surd.

Example: $\frac{3}{\sqrt{7}} = \frac{3 \times \sqrt{7}}{\sqrt{7} \times \sqrt{7}} = \frac{3\sqrt{7}}{7}$

Type 2: Denominator of the form $x + \sqrt{y}$ or $x - \sqrt{y}$

To rationalise these, multiply the numerator and denominator by the conjugate (the same expression but with the opposite sign). This creates a difference of two squares in the denominator: $(x + \sqrt{y})(x - \sqrt{y}) = x^2 - y$.

Example: Rationalise $\frac{5}{3 + 2\sqrt{5}}$. Multiply top and bottom by $3 - 2\sqrt{5}$:

$\frac{5(3 - 2\sqrt{5})}{(3 + 2\sqrt{5})(3 - 2\sqrt{5})} = \frac{5(3 - 2\sqrt{5})}{3^2 - (2\sqrt{5})^2} = \frac{5(3 - 2\sqrt{5})}{9 - 20}$

This simplifies to: $\frac{5(3 - 2\sqrt{5})}{-11} = \frac{5(2\sqrt{5} - 3)}{11}$

Example: Rationalise $\frac{2+\sqrt{3}}{2-\sqrt{3}}$. Multiply top and bottom by $2 + \sqrt{3}$:

$\frac{(2+\sqrt{3})(2+\sqrt{3})}{(2-\sqrt{3})(2+\sqrt{3})} = \frac{(2+\sqrt{3})^2}{4-3} = \frac{4 + 4\sqrt{3} + 3}{1} = 7 + 4\sqrt{3}$

Type 3: Denominator of the form $\sqrt{x} - \sqrt{y}$

Multiply by the conjugate $\sqrt{x} + \sqrt{y}$ to yield the difference of squares $(\sqrt{x} - \sqrt{y})(\sqrt{x} + \sqrt{y}) = x - y$.

Example: Rationalise $\frac{3}{\sqrt{5} - \sqrt{2}}$. Multiply top and bottom by $\sqrt{5} + \sqrt{2}$:

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

Complex Example: Multiple Factors

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

Multiply top and bottom by $\sqrt{15}(\sqrt{7} + \sqrt{2})$:

$\frac{\sqrt{5}\sqrt{15}(\sqrt{7} + \sqrt{2})}{\sqrt{15}\sqrt{15}(\sqrt{7} - \sqrt{2})(\sqrt{7} + \sqrt{2})} = \frac{\sqrt{75}(\sqrt{7} + \sqrt{2})}{15(7 - 2)}$

Simplify $\sqrt{75}$ as $\sqrt{25}\sqrt{3} = 5\sqrt{3}$:

$\frac{5\sqrt{3}(\sqrt{7} + \sqrt{2})}{15 \times 5} = \frac{\sqrt{3}(\sqrt{7} + \sqrt{2})}{15}$