Basic Algebraic Manipulation and Indices

Collecting Like Terms

Every time you tidy your room, you naturally group similar items together—books with books, clothes with clothes; in algebra, we do exactly the same thing.

An expression is a collection of terms without an equals sign.
A single term is a number, variable, or product of numbers and variables (e.g., $3x$ or $4y^2$ ).
A variable is a letter used to represent an unknown value.
The number in front of a variable is called the coefficient. If no number is written (e.g., $x$ or $-y$ ), the coefficient is implicitly $1$ or $-1$ .
Like terms are terms that contain the exact same variables raised to the exact same powers.
The process of combining these terms is called simplification. When adding or subtracting, you change the coefficient, never the power.
The plus ( $+$ ) or minus ( $-$ ) sign immediately to the left of a term belongs to that term.

Worked Example:

Simplify the expression: $x + 4x^2 + 3x - 5x^2$

Step 1: Group the coefficients of the $x$ terms (remember $x$ implicitly has a coefficient of $1$ ).

$(1 + 3)x = 4x$

Step 2: Group the coefficients of the $x^2$ terms.

$(4 - 5)x^2 = -1x^2 = -x^2$

Step 3: Write the final simplified expression.

$4x - x^2$

Expanding Single Brackets

You can think of a bracket as a sealed box of items, and a number outside tells you how many of those boxes you have.

To expand a bracket, you remove the brackets by multiplying the external term by every single term on the inside.
This uses the distributive law, which states that multiplying a sum by a number gives the same product as multiplying each addend separately:

a(b + c) = ab + ac

When expanding, pay close attention to signs: multiplying a negative by a positive gives a negative result.
If a question asks you to "Expand and simplify", you must expand the brackets first, and then collect any like terms.

Worked Example:

Expand: $3x(x + 2)$

Step 1: Substitute the terms into the distributive law formula $a(b + c) = ab + ac$ .

$3x(x + 2) = (3x)(x) + (3x)(2)$

Step 2: Calculate each separate product.

$(3x)(x) = 3x^2$
$(3x)(2) = 6x$

Step 3: Combine to form the expanded expression.

$3x^2 + 6x$

Factorising Expressions

You can snap a piece of chalk, but try snapping a diamond; similarly, breaking down algebraic expressions requires finding the core components they are built from.

Factorisation is the process of writing an expression as a product of factors, usually by introducing brackets. It is the inverse operation to expanding.
You must test the divisibility of the terms to find the Highest Common Factor (HCF).
The HCF includes both the largest numerical factor dividing all coefficients and the lowest power of any variable common to every term.
In Edexcel exams, the command "Factorise fully" means the expression left inside the bracket must not share any more common factors.

Worked Example:

Factorise fully: $12x^2 + 18x$

Step 1: Identify the HCF of the numerical coefficients.

The largest number that divides $12$ and $18$ is $6$ .

Step 2: Identify the HCF of the algebraic variables.

Both terms contain an $x$ , and the lowest power is $x^1$ (or just $x$ ).
Overall HCF = $6x$ .

Step 3: Divide the original terms by the HCF to find what goes inside the bracket.

$12x^2 \div 6x = 2x$
$18x \div 6x = 3$

Step 4: Write the final factorised expression.

$6x(2x + 3)$

Index Laws: Multiplication and Division

Writing out $y \times y \times y \times y \times y \times y \times y$ is incredibly tedious, which is why index notation was invented to keep track of repeated multiplication.

The base is the number or variable being multiplied by itself.
The index (also called an exponent or power) indicates how many times the base is multiplied by itself.
These rules are known as index laws.
The Multiplication Law: When multiplying terms with the same base, add the indices.

a^m \times a^n = a^{m+n}

The Division Law: When dividing terms with the same base, subtract the indices. If subtracting results in a negative index, it represents a reciprocal (e.g., $x^{-3} = \frac{1}{x^3}$ ).

a^m \div a^n = a^{m-n}

Note that these index laws only apply when the bases are identical. Numerical coefficients are multiplied or divided normally.

Worked Example:

Simplify: $3y^3 \times 2y^4$

Step 1: Multiply the numerical coefficients.

$3 \times 2 = 6$

Step 2: Apply the multiplication law to the variables by adding the indices.

$y^{3+4} = y^7$

Step 3: Combine for the final answer.

$6y^7$

Simplify: $\frac{12x^5y^2}{4x^2y}$

Step 1: Divide the numerical coefficients.

$12 \div 4 = 3$

Step 2: Apply the division law to the $x$ terms by subtracting the indices.

$x^{5-2} = x^3$

Step 3: Apply the division law to the $y$ terms (remember $y$ is $y^1$ ).

$y^{2-1} = y^1 = y$

Step 4: Combine for the final answer.

$3x^3y$

The Power of a Power Rule

Why does a small power outside a bracket have such a massive impact on the final value?

The power of a power rule is used when a term already raised to a power is raised to another power. In this case, you multiply the indices:

(a^m)^n = a^{mn}

Crucially, if there is a numerical coefficient inside the bracket, it must be raised to the external power normally.

Worked Example:

Simplify: $(3y^2)^3$

Step 1: Raise the numerical coefficient to the outside power.

$3^3 = 27$

Step 2: Apply the power of a power rule to the variable by multiplying the indices.

$y^{2 \times 3} = y^6$

Step 3: Combine for the final answer.

$27y^6$

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Get full access to revision notes, key terms, and exam tips for every subtopic.

Exam Tips

Common Mistake
Students often forget to include the sign to the left of a term when collecting like terms. For example, in $5x - 3 + 2x$ , they incorrectly calculate $3 + 2x$ instead of grouping $5x + 2x$ .
Common Mistake
When expanding single brackets, students frequently forget to multiply the second term inside the bracket by the term on the outside (e.g., writing $3(x + 4)$ as $3x + 4$ instead of $3x + 12$ ).
3
In 'Expand and simplify' questions worth 2 marks, examiners will usually award 1 method mark for correctly expanding the brackets, so always write out the unsimplified expansion before collecting like terms.
Common Mistake
When applying the power of a power rule to a term with a coefficient like $(2x^5)^3$ , students often forget to raise the numerical coefficient to the power, incorrectly writing $2x^{15}$ instead of $8x^{15}$ .

Key Terms(21)

Expression: A collection of algebraic terms without an equals sign.
Term: A single number, variable, or product of numbers and variables separated by plus or minus signs in an algebraic expression.
Variable: A letter used to represent an unknown value or a range of values.
Coefficient: The numerical multiplier of a variable in an algebraic term.
Like terms: Terms that contain the same variables raised to the identical powers.
Simplification: The process of combining like terms to make an expression shorter or simpler.
Expand: To remove brackets by multiplying the external term by every term inside.
Brackets: Symbols used to group terms together, indicating that operations inside should be treated as a single quantity.
Distributive law: The mathematical rule stating that multiplying a sum by a number gives the same result as multiplying each addend separately.
Product: The result obtained by multiplying two or more quantities together.
Factorisation: The process of writing an algebraic expression as a product of its factors, typically by taking out common factors and using brackets.
Inverse operation: An operation that reverses the effect of another operation (e.g., expanding and factorising).
Divisibility: The capacity of a number or algebraic term to be divided by another without leaving a remainder.
Highest Common Factor (HCF): The largest factor common to all terms in an algebraic expression, including both numbers and variables.
Base: The number or variable that is being repeatedly multiplied by itself in an index expression.
Index: A number indicating how many times the base is multiplied by itself.
Exponent: Another word for index or power.
Power: Another word for index or exponent.
Index laws: The set of mathematical rules governing operations with powers and indices.
Reciprocal: The multiplicative inverse of a value, which in index laws is generated by a negative exponent.
Power of a power rule: The index law stating that when a term with a power is raised to another power, the indices are multiplied.

Previous Subtopic: Algebraic VocabularyAlgebraic Vocabulary Next NoteExpanding and Factorising Quadratics

Back to Expression Simplification

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Key Terms(21)

Expression: A collection of algebraic terms without an equals sign.
Term: A single number, variable, or product of numbers and variables separated by plus or minus signs in an algebraic expression.
Variable: A letter used to represent an unknown value or a range of values.
Coefficient: The numerical multiplier of a variable in an algebraic term.
Like terms: Terms that contain the same variables raised to the identical powers.
Simplification: The process of combining like terms to make an expression shorter or simpler.
Expand: To remove brackets by multiplying the external term by every term inside.
Brackets: Symbols used to group terms together, indicating that operations inside should be treated as a single quantity.
Distributive law: The mathematical rule stating that multiplying a sum by a number gives the same result as multiplying each addend separately.
Product: The result obtained by multiplying two or more quantities together.
Factorisation: The process of writing an algebraic expression as a product of its factors, typically by taking out common factors and using brackets.
Inverse operation: An operation that reverses the effect of another operation (e.g., expanding and factorising).
Divisibility: The capacity of a number or algebraic term to be divided by another without leaving a remainder.
Highest Common Factor (HCF): The largest factor common to all terms in an algebraic expression, including both numbers and variables.
Base: The number or variable that is being repeatedly multiplied by itself in an index expression.
Index: A number indicating how many times the base is multiplied by itself.
Exponent: Another word for index or power.
Power: Another word for index or exponent.
Index laws: The set of mathematical rules governing operations with powers and indices.
Reciprocal: The multiplicative inverse of a value, which in index laws is generated by a negative exponent.
Power of a power rule: The index law stating that when a term with a power is raised to another power, the indices are multiplied.

Basic Algebraic Manipulation and Indices

Collecting Like Terms

Every time you tidy your room, you naturally group similar items together—books with books, clothes with clothes; in algebra, we do exactly the same thing.

An expression is a collection of terms without an equals sign.
A single term is a number, variable, or product of numbers and variables (e.g., $3x$ or $4y^2$ ).
A variable is a letter used to represent an unknown value.
The number in front of a variable is called the coefficient. If no number is written (e.g., $x$ or $-y$ ), the coefficient is implicitly $1$ or $-1$ .
Like terms are terms that contain the exact same variables raised to the exact same powers.
The process of combining these terms is called simplification. When adding or subtracting, you change the coefficient, never the power.
The plus ( $+$ ) or minus ( $-$ ) sign immediately to the left of a term belongs to that term.

Worked Example:

Simplify the expression: $x + 4x^2 + 3x - 5x^2$

Step 1: Group the coefficients of the $x$ terms (remember $x$ implicitly has a coefficient of $1$ ).

$(1 + 3)x = 4x$

Step 2: Group the coefficients of the $x^2$ terms.

$(4 - 5)x^2 = -1x^2 = -x^2$

Step 3: Write the final simplified expression.

$4x - x^2$

Expanding Single Brackets

You can think of a bracket as a sealed box of items, and a number outside tells you how many of those boxes you have.

To expand a bracket, you remove the brackets by multiplying the external term by every single term on the inside.
This uses the distributive law, which states that multiplying a sum by a number gives the same product as multiplying each addend separately:

a(b + c) = ab + ac

When expanding, pay close attention to signs: multiplying a negative by a positive gives a negative result.
If a question asks you to "Expand and simplify", you must expand the brackets first, and then collect any like terms.

Worked Example:

Expand: $3x(x + 2)$

Step 1: Substitute the terms into the distributive law formula $a(b + c) = ab + ac$ .

$3x(x + 2) = (3x)(x) + (3x)(2)$

Step 2: Calculate each separate product.

$(3x)(x) = 3x^2$
$(3x)(2) = 6x$

Step 3: Combine to form the expanded expression.

$3x^2 + 6x$

Factorising Expressions

You can snap a piece of chalk, but try snapping a diamond; similarly, breaking down algebraic expressions requires finding the core components they are built from.

Factorisation is the process of writing an expression as a product of factors, usually by introducing brackets. It is the inverse operation to expanding.
You must test the divisibility of the terms to find the Highest Common Factor (HCF).
The HCF includes both the largest numerical factor dividing all coefficients and the lowest power of any variable common to every term.
In Edexcel exams, the command "Factorise fully" means the expression left inside the bracket must not share any more common factors.

Worked Example:

Factorise fully: $12x^2 + 18x$

Step 1: Identify the HCF of the numerical coefficients.

The largest number that divides $12$ and $18$ is $6$ .

Step 2: Identify the HCF of the algebraic variables.

Both terms contain an $x$ , and the lowest power is $x^1$ (or just $x$ ).
Overall HCF = $6x$ .

Step 3: Divide the original terms by the HCF to find what goes inside the bracket.

$12x^2 \div 6x = 2x$
$18x \div 6x = 3$

Step 4: Write the final factorised expression.

$6x(2x + 3)$

Index Laws: Multiplication and Division

Writing out $y \times y \times y \times y \times y \times y \times y$ is incredibly tedious, which is why index notation was invented to keep track of repeated multiplication.

The base is the number or variable being multiplied by itself.
The index (also called an exponent or power) indicates how many times the base is multiplied by itself.
These rules are known as index laws.
The Multiplication Law: When multiplying terms with the same base, add the indices.

a^m \times a^n = a^{m+n}

The Division Law: When dividing terms with the same base, subtract the indices. If subtracting results in a negative index, it represents a reciprocal (e.g., $x^{-3} = \frac{1}{x^3}$ ).

a^m \div a^n = a^{m-n}

Note that these index laws only apply when the bases are identical. Numerical coefficients are multiplied or divided normally.

Worked Example:

Simplify: $3y^3 \times 2y^4$

Step 1: Multiply the numerical coefficients.

$3 \times 2 = 6$

Step 2: Apply the multiplication law to the variables by adding the indices.

$y^{3+4} = y^7$

Step 3: Combine for the final answer.

$6y^7$

Simplify: $\frac{12x^5y^2}{4x^2y}$

Step 1: Divide the numerical coefficients.

$12 \div 4 = 3$

Step 2: Apply the division law to the $x$ terms by subtracting the indices.

$x^{5-2} = x^3$

Step 3: Apply the division law to the $y$ terms (remember $y$ is $y^1$ ).

$y^{2-1} = y^1 = y$

Step 4: Combine for the final answer.

$3x^3y$

The Power of a Power Rule

Why does a small power outside a bracket have such a massive impact on the final value?

The power of a power rule is used when a term already raised to a power is raised to another power. In this case, you multiply the indices:

(a^m)^n = a^{mn}

Crucially, if there is a numerical coefficient inside the bracket, it must be raised to the external power normally.

Worked Example:

Simplify: $(3y^2)^3$

Step 1: Raise the numerical coefficient to the outside power.

$3^3 = 27$

Step 2: Apply the power of a power rule to the variable by multiplying the indices.

$y^{2 \times 3} = y^6$

Step 3: Combine for the final answer.

$27y^6$

Sign up to continue reading

Get full access to revision notes, key terms, and exam tips for every subtopic.

Exam Tips

Common Mistake
Students often forget to include the sign to the left of a term when collecting like terms. For example, in $5x - 3 + 2x$ , they incorrectly calculate $3 + 2x$ instead of grouping $5x + 2x$ .
Common Mistake
When expanding single brackets, students frequently forget to multiply the second term inside the bracket by the term on the outside (e.g., writing $3(x + 4)$ as $3x + 4$ instead of $3x + 12$ ).
3
In 'Expand and simplify' questions worth 2 marks, examiners will usually award 1 method mark for correctly expanding the brackets, so always write out the unsimplified expansion before collecting like terms.
Common Mistake
When applying the power of a power rule to a term with a coefficient like $(2x^5)^3$ , students often forget to raise the numerical coefficient to the power, incorrectly writing $2x^{15}$ instead of $8x^{15}$ .

Key Terms(21)

Expression: A collection of algebraic terms without an equals sign.
Term: A single number, variable, or product of numbers and variables separated by plus or minus signs in an algebraic expression.
Variable: A letter used to represent an unknown value or a range of values.
Coefficient: The numerical multiplier of a variable in an algebraic term.
Like terms: Terms that contain the same variables raised to the identical powers.
Simplification: The process of combining like terms to make an expression shorter or simpler.
Expand: To remove brackets by multiplying the external term by every term inside.
Brackets: Symbols used to group terms together, indicating that operations inside should be treated as a single quantity.
Distributive law: The mathematical rule stating that multiplying a sum by a number gives the same result as multiplying each addend separately.
Product: The result obtained by multiplying two or more quantities together.
Factorisation: The process of writing an algebraic expression as a product of its factors, typically by taking out common factors and using brackets.
Inverse operation: An operation that reverses the effect of another operation (e.g., expanding and factorising).
Divisibility: The capacity of a number or algebraic term to be divided by another without leaving a remainder.
Highest Common Factor (HCF): The largest factor common to all terms in an algebraic expression, including both numbers and variables.
Base: The number or variable that is being repeatedly multiplied by itself in an index expression.
Index: A number indicating how many times the base is multiplied by itself.
Exponent: Another word for index or power.
Power: Another word for index or exponent.
Index laws: The set of mathematical rules governing operations with powers and indices.
Reciprocal: The multiplicative inverse of a value, which in index laws is generated by a negative exponent.
Power of a power rule: The index law stating that when a term with a power is raised to another power, the indices are multiplied.

Previous Subtopic: Algebraic VocabularyAlgebraic Vocabulary Next NoteExpanding and Factorising Quadratics

Back to Expression Simplification

Put your knowledge into practice — try past paper questions for Mathematics

Key Terms(21)

Expression: A collection of algebraic terms without an equals sign.
Term: A single number, variable, or product of numbers and variables separated by plus or minus signs in an algebraic expression.
Variable: A letter used to represent an unknown value or a range of values.
Coefficient: The numerical multiplier of a variable in an algebraic term.
Like terms: Terms that contain the same variables raised to the identical powers.
Simplification: The process of combining like terms to make an expression shorter or simpler.
Expand: To remove brackets by multiplying the external term by every term inside.
Brackets: Symbols used to group terms together, indicating that operations inside should be treated as a single quantity.
Distributive law: The mathematical rule stating that multiplying a sum by a number gives the same result as multiplying each addend separately.
Product: The result obtained by multiplying two or more quantities together.
Factorisation: The process of writing an algebraic expression as a product of its factors, typically by taking out common factors and using brackets.
Inverse operation: An operation that reverses the effect of another operation (e.g., expanding and factorising).
Divisibility: The capacity of a number or algebraic term to be divided by another without leaving a remainder.
Highest Common Factor (HCF): The largest factor common to all terms in an algebraic expression, including both numbers and variables.
Base: The number or variable that is being repeatedly multiplied by itself in an index expression.
Index: A number indicating how many times the base is multiplied by itself.
Exponent: Another word for index or power.
Power: Another word for index or exponent.
Index laws: The set of mathematical rules governing operations with powers and indices.
Reciprocal: The multiplicative inverse of a value, which in index laws is generated by a negative exponent.
Power of a power rule: The index law stating that when a term with a power is raised to another power, the indices are multiplied.