Simplifying Algebraic Expressions
Simplifying algebraic expressions is the first real skill in IGCSE algebra: you collect like terms, then multiply, divide and take powers of single terms using the basic index laws. This Edexcel IGCSE Maths page gives you clear worked examples, a colour-coded method for collecting like terms, and the index rules you need, all in exam style. When you are ready, scroll down to the Practice Questions: a random generator with auto-marked answers and instant feedback, so you can build speed and accuracy fast.
What is an algebraic expression?
An algebraic expression combines numbers, letters (variables) and operations such as \(+\), \(-\), \(\times\) and division. An expression has no equals sign, so you cannot solve it, only simplify it: rewrite it in the shortest, tidiest form that means the same thing.
- Variables are letters that stand for numbers, for example \(x\), \(y\) or \(p\).
- Constants are ordinary fixed numbers, for example \(5\) or \(-3\).
- Like terms have exactly the same letter part, for example \(3x\) and \(7x\), or \(2x^2y\) and \(5x^2y\).
How to simplify by collecting like terms:
- If there are brackets, expand them first.
- Group the terms of each family together, keeping each sign with its term.
- Add or subtract the number in front (the coefficient) of each family; the letter part does not change.
Collecting like terms, step by step
Colour marks each family, so you can see exactly what combines with what. Only terms of the same colour can be added or subtracted.
The \(x\) terms combine to \(4x\) and the \(y\) terms combine to \(5y\). They are different families, so the answer keeps both.
Index laws for simplifying
When you multiply, divide or take a power of terms with the same base, three laws do the work. These are the basics for positive whole-number powers. For zero, negative and fractional powers, see the full Laws of Indices page.
Multiply: add the powers
\( a^m \times a^n = a^{\,m+n} \)
\[ \begin{array}{rcl} x^2 \times x^3 &=& x^{\,2+3} \\ &=& x^5 \end{array} \]Divide: subtract the powers
\( \dfrac{a^m}{a^n} = a^{\,m-n} \)
\[ \begin{array}{rcl} \dfrac{x^6}{x^2} &=& x^{\,6-2} \\ &=& x^4 \end{array} \]Power of a power: multiply
\( (a^m)^n = a^{\,mn} \)
\[ \begin{array}{rcl} (x^3)^2 &=& x^{\,3\times 2} \\ &=& x^6 \end{array} \]Worked examples
💡 Example 1: collect like terms
Simplify \(\;6a + 5 - 2a + 3 - a\).
What is happening?
Group the \(a\) terms and the constants. The coefficients give \(6-2-1=3\), so the \(a\) family is \(3a\); the constants give \(5+3=8\).
💡 Example 2: different powers are different families
Simplify \(\;4x^2 + 3x - x^2 + 5x\).
What is happening?
\(x^2\) and \(x\) are separate families, so they never combine. Collect each one on its own: the \(x^2\) family is \(3x^2\) and the \(x\) family is \(8x\).
💡 Example 3: multiply and divide terms
Multiply \(\;4p \times 3p^2\).
Divide \(\;\dfrac{20x^5}{4x^2}\).
What is happening?
Deal with the numbers and the letters separately. To multiply, multiply the numbers and add the powers; to divide, divide the numbers and subtract the powers.
💡 Example 4: power of a term
Simplify \(\;(2x^3)^2\).
What is happening?
The power outside the bracket applies to everything inside: the number is squared to give \(2^2=4\), and the power of \(x\) is multiplied, \((x^3)^2=x^6\).
🔑 Key Points
- Only add or subtract like terms: same letters, same powers.
- Add or subtract the coefficients; the letter part stays the same.
- Multiplying terms: multiply the numbers and add the powers of each letter.
- Dividing terms: divide the numbers and subtract the powers.
- Power of a term: raise the number to that power and multiply the powers.
- Give the final answer fully simplified, with each family written once.
⚠️ Common Pitfalls
- Mixing families: \(x\) and \(x^2\) are different, so \(3x+2x^2\) does not simplify.
- Losing a sign: keep the sign attached to its term when you regroup.
- Subtracting a bracket flips every sign inside it, not just the first.
- Adding powers when collecting: \(3x+2x=5x\), never \(5x^2\).
- Forgetting the coefficient in a power: \((2x^3)^2=4x^6\), not \(2x^6\).