Algebraic Expressions

By Stage 9 you already know that to simplify an expression you collect like terms — terms with exactly the same letter part. What changes now is the level: expressions are longer, signs are mixed, and powers appear everywhere, so a steady method matters.

Take $7x^2 - 4x + 2 - 3x^2 + 9x - 5$. Sort it carefully into groups, keeping each sign attached:

So the expression simplifies to $4x^2 + 5x - 3$.

A few habits keep you accurate:

• Treat the sign in front of a term as part of that term.
• A lone $-x$ has a coefficient of $-1$.
• $x$ and $x^2$ are never like terms — keep their groups apart.
• Write the simplified answer in a sensible order: highest power first.

When you multiply terms, deal with the numbers first, then the letters. For $4a^2 \times 3a$: multiply the numbers $4 \times 3 = 12$, then the letters $a^2 \times a = a^3$ (because $a^2 \times a$ stacks three $a$s together). The result is $12a^3$.

Dividing is easiest written as a fraction. For $\tfrac{15x^5}{3x^2}$: divide the numbers $15 \div 3 = 5$, then subtract the indices $x^5 \div x^2 = x^3$. The answer is $5x^3$.

A few more to fix the pattern:

• $2x \times 5x = 10x^2$ — numbers give 10, indices add to give $x^2$.
• $\tfrac{12a^4}{4a} = 3a^3$ — numbers give 3, indices subtract to give $a^3$.
• $6p^3 \times 2p^2 = 12p^5$.

The rule underneath both is the index law: multiplying adds indices, dividing subtracts them. Get this and term arithmetic becomes quick and reliable.

An index (or power) tells you how many times a letter is multiplied by itself. At Stage 9 the index laws become a proper set of rules you apply to algebraic terms.

Here is each law with an algebra example:

• Multiplying: $x^m \times x^n = x^{m+n}$. So $x^4 \times x^3 = x^7$.
• Dividing: $x^m \div x^n = x^{m-n}$. So $x^7 \div x^3 = x^4$.
• Power of a power: $(x^m)^n = x^{mn}$. So $(x^3)^2 = x^6$ — six $x$s in total.
• Zero index: $x^0 = 1$. This follows from dividing: $x^3 \div x^3 = x^0$, and anything divided by itself is 1.

These laws only work when the base (the letter) is the same. $x^3 \times y^2$ cannot be combined — the bases differ. Keep an eye on the base and the rules never let you down.

Sometimes a whole term is raised to a power, like $(3x^2)^3$. The key idea: the power applies to everything inside the bracket — the number and the letter.

So $(3x^2)^3 = 3^3 \times (x^2)^3 = 27x^6$.

Work through each part:

• The number: $3^3 = 3 \times 3 \times 3 = 27$.
• The letter: $(x^2)^3 = x^{2 \times 3} = x^6$, using the power-of-a-power law.

A few more:

• $(2y)^4 = 2^4 \times y^4 = 16y^4$.
• $(5a^3)^2 = 25a^6$.
• $(x^2 y)^3 = x^6 y^3$ — every letter inside gets raised.

The single most common slip here is raising the letter but leaving the number alone — writing $(3x^2)^3 = 3x^6$ instead of $27x^6$. The power touches everything inside.

To expand a bracket you multiply every term inside by the term outside. At Stage 9 the outside term often carries a power, so your index skills come straight in.

So $2x^2(3x + 4) = 6x^3 + 8x^2$.

Most Stage 9 questions have two brackets to expand and then tidy. Take $3(2x + 5) - 2(x - 4)$:

1. Expand the first bracket: $3(2x + 5) = 6x + 15$.
2. Expand the second, minding the minus: $-2(x - 4) = -2x + 8$.
3. Collect like terms: $6x - 2x = 4x$ and $15 + 8 = 23$.

The result is $4x + 23$. The golden rule never changes: expand fully first, then collect like terms in one careful sweep.

Sharpening your expression and index skills now opens up the rest of algebra:

• Expansion and factorisation rely on confident multiplying of terms, including the product of two brackets.
• Equations and inequalities are solved much faster when you can simplify both sides first.
• Standard form at IGCSE uses the index laws directly with powers of 10.
• Formulae are rearranged and simplified using these exact moves.

In everyday life, index notation models growth and scale — like the number of possible passwords, or how a quantity doubles again and again.

If algebra ever feels hard later, it is often an index law slip or a missed like-term underneath. Come back to this guide whenever you need a refresher — there is no prize for rushing.