Algebraic Fractions, Indices, and Equations

Section 1Algebraic fractions

An algebraic fraction has variables in the numerator, denominator, or both. Simplify by factorising first, then cancelling common factors. Watch out: you cancel factors, not random terms, so (x + 3) / x cannot simplify to 3.

Section 2Restrictions

A denominator cannot be zero. If a fraction has denominator x - 4, then x cannot equal 4. This matters because a cancelled factor can still create a restriction from the original expression.

Section 3Laws of indices

Index laws help you simplify powers with the same base. When multiplying powers with the same base, add the indices. When dividing powers with the same base, subtract the indices. When raising a power to another power, multiply the indices.

Section 4Zero and negative indices

Any non-zero number to the power 0 is 1. A negative index means reciprocal, so x^-2 = 1 / x^2. Easy to mix up - a negative index does not make the value negative; it sends the power to the denominator.

Section 5Linear equations

A linear equation has the variable to the first power, such as 3x + 5 = 20. Solve by undoing operations in a balanced way. In plain words, whatever you do to one side, do to the other side.

Section 6Linear inequalities

A linear inequality uses signs like <, >, <=, or >= instead of equals. Solve it like an equation, but remember the key trap: when you multiply or divide both sides by a negative number, reverse the inequality sign.

Section 7Changing the subject

Changing the subject means rearranging a formula so a different variable stands alone. Use inverse operations step by step. Here's the trick: treat the formula like an equation and isolate the new subject carefully, especially when the variable appears inside brackets or a denominator.