Algebraic Expressions

An algebraic expression is a mix of numbers, letters and operations — for example $5x + 3y - 7$. There is no equals sign, so it is not something you solve; it is something you read, tidy and sometimes stretch out.

The pieces between the $+$ or $-$ signs are called terms. So $5x + 3y - 7$ has three terms: $5x$, $+3y$ and $-7$. Each letter term is built from a coefficient (the number in front) and a variable (the letter). In $5x$ the coefficient is $5$ and the variable is $x$.

A few quiet conventions worth knowing: $1x$ is always written as $x$, $-1x$ as $-x$, and $x \times y$ as $xy$. A term with no letter — like the $-7$ above — is called a constant. Read every expression slowly the first time so you do not lose track of a sign.

Like terms are terms that have exactly the same letter part. So $5x$ and $2x$ are like terms, and so are $3xy$ and $-4xy$. But $5x$ and $5x^2$ are not like terms — the powers are different — and neither are $5x$ and $5y$.

To simplify, you add or subtract the coefficients of like terms and leave the letter part untouched.

• $5x + 2x = 7x$
• $9y - 4y = 5y$
• $7a + 3b - 2a + 5b = 5a + 8b$

Two tips speed this up. First, carry each sign with its term when you regroup — $7a - 2a$, not $7a + 2a$. Second, write the simplified terms in a sensible order: letters in alphabetical order, then the constant last, so $3a + 4b - 2$ reads cleanly.

To expand (or multiply out) a bracket, multiply every term inside by the factor sitting outside. This works because of the distributive law: $a(b + c) = ab + ac$.

• $3(x + 4) = 3x + 12$
• $-2(y - 5) = -2y + 10$
• $x(x + 7) = x^2 + 7x$

After expanding, collect like terms if any new ones appear. For example $4(x + 3) + 2(x - 5) = 4x + 12 + 2x - 10 = 6x + 2$. Expand first, then tidy — one job at a time, and the signs stay under control.

An index (or power) tells you how many times a base is multiplied by itself, so $x^4 = x \times x \times x \times x$. Three rules cover almost everything you need.

• Multiplying with the same base — add the indices: $x^a \times x^b = x^{a+b}$. So $x^5 \times x^2 = x^7$.
• Dividing with the same base — subtract the indices: $\dfrac{x^a}{x^b} = x^{a-b}$. So $\dfrac{x^7}{x^3} = x^4$.
• Power of a power — multiply the indices: $(x^a)^b = x^{ab}$. So $(x^3)^2 = x^6$.

Two extra facts close the gaps. Anything (except zero) to the power $0$ is $1$, so $x^0 = 1$. And a number with no index shown has an invisible index of $1$, so $x = x^1$. When a term has both a coefficient and a variable — $3x^2 \times 4x^5$ — multiply the coefficients separately from the indices: $3 \times 4 = 12$, $x^{2+5} = x^7$, giving $12x^7$.

A typical "simplify" question pulls in several tools at once. A reliable order is:

1. Expand any brackets first.
2. Apply the index laws to combine same-base powers.
3. Collect like terms to finish.

Take $4(2x + 3) - 2(x - 5)$. Expand both brackets carefully: $8x + 12 - 2x + 10$. There are no powers to combine, so collect: $(8x - 2x) + (12 + 10) = 6x + 22$.

Try one with powers: $2x \times 3x^2 + 5x^3 - x \times x^2$. Use the index law on each product first: $2x \times 3x^2 = 6x^3$ and $x \times x^2 = x^3$. The expression becomes $6x^3 + 5x^3 - x^3 = 10x^3$.

Two final habits keep you accurate. Write one step per line — squashing two steps together is where sign errors hide. And sense-check the answer — if the original had only $x$ terms and constants, the simplified version should too.

Fluent expression-work unlocks a lot of the maths that comes next:

• Solving equations starts with simplifying both sides — collect like terms, expand brackets, then balance.
• Formulae use exactly the same rules, just with more letters: substituting and rearranging both need confident algebra.
• Quadratic expressions build directly on expanding and the index laws ($x \times x = x^2$).
• Graphs and sequences describe patterns with expressions, so tidy notation pays off there too.

If a later topic feels slow, it is often a like-terms slip or a missed sign that needs a quick refresh. Come back to this guide whenever you need a steady reset — accuracy beats speed every time.