Key points about laws of indices
The notation in which a product such as 𝑛 × 𝑛 × 𝑛 × 𝑛 is recorded as 𝑛⁴. In this example the number 4 is called the index (plural indices) and the number represented by 𝑛 is the base. is when a The number or variable that is raised to a power when using index notation. is raised to a A number that is multiplied by itself one or more times is raised to a power. 𝑛⁴ is ‘𝑛 to the power of 4’ and means 𝑛 × 𝑛 × 𝑛 × 𝑛. The power is the value of the index or exponent., to condense a calculation.
The Agreed rules for simplifying expressions (involving multiplication, division and raising to a power) using index notation with the same base. simplify:
- Multiplying with indices.
- Dividing with indices.
- Raising a power to another power.
Refresh your knowledge with these guides on powers and roots.
Video – Laws of indices
Watch this video to learn about the three laws of indices relating to multiplying, dividing and raising a power to a power.
Laws of indices: Multiplication, division, and power to a power.
A base number or variable raised to an index, or power, means a number has been multiplied by itself that many times.
When working with indices with the same base, there are laws you can follow to make calculations easier.
Indices is the plural of index.
When multiplying numbers with the same base, you add the indices, or powers, together.
For example, x to the power of 4, multiplied by x to the power of 3, is x to the power of 4 add 3, or x to the power of 7.
When dividing numbers with the same base, you subtract the powers.
For example: x to the 6, divided by x to the 2 is equal to x to the power of 6 subtract 2, or x to the 4.
And finally, when raising a power to a power, you multiply the powers.
So, x to the 5, all to the power of 4, is x to the power of 5 times 4, which is x to the 20.
Let's look at an example:
Simplify p to the 3, multiplied by p to the 4, all over p to the 2.
Use the law for multiplication on the numerator to get p to the power of 3 add 4, which is p to the 7.
Then use the law for division to divide this by the denominator: p to the power of 7 subtract 2 gives a final answer of p to the 5.
Here's another example:
Simplify 2x squared, all to the power of 3.
Pause the video to have a go.
Be careful here!
The x has a coefficient of 2. The power of 3 applies to everything inside the brackets, so you can rewrite this as 2 to the power of 3, multiplied by x squared, all to the power of 3.
2 to the 3 is 8, and then using the law for raising a power to a power, x squared, all to the power of 3, gives x to the power of 2 times 3. Which is x to the 6.
Simplified, the answer is 8x to the 6.
Check your understanding
How to use an index or power
Index notation involves a The number or variable that is raised to a power when using index notation. number (or variable) raised to a A number that is multiplied by itself one or more times is raised to a power. 𝑛⁴ is ‘𝑛 to the power of 4’ and means 𝑛 × 𝑛 × 𝑛 × 𝑛. The power is the value of the index or exponent..
Follow the examples and working out below.
- Read the term as the base to the power of the Positioned above and to the right of a number. It is an abbreviation of repeated multiplication, eg 7³ means 7 × 7 × 7. An index may be positive, negative, zero or fractional. Indices are also referred to as powers or exponents.:
- 𝑦⁹ is ‘𝑦 to the power of nine’.
- For a positive Integers are numbers with no fraction or decimal part. They can be positive, negative or zero. 42, –8, and 10,000 are examples of integers. index, the term represents how many times to use the base in a multiplication.
- A numerical term can be evaluated by working out the repeated multiplication.
- A value can be To replace a variable with a number or an expression. for a variable to To find the value of a numerical or an algebraic expression. an algebraic term.
GCSE exam-style questions
- Find the value of 𝑥 when 3\(^𝑥\) = 81.
𝑥 = 4
If 3\(^𝑥\) = 81, find out how many times 3 needs to be multiplied by itself to get the answer 81:
- 3 × 3 = 9
- 3 × 3 × 3 = 27
- 3 × 3 × 3 × 3 = 81
3 is multiplied by itself four times.
The index is 4.
- 3⁴ = 81
- Which is greater, 2⁶ or 6²?
2⁶ > 6²
- 2⁶ is 2 × 2 × 2 × 2 × 2 × 2 = 64
- 6² is 6 × 6 = 36
64 is greater than 36, so 2⁶ > 6².
How to multiply indices
The Agreed rules for simplifying expressions (involving multiplication, division and raising to a power) using index notation with the same base. are used when the The number or variable that is raised to a power when using index notation. is the same.
Follow the examples and working out below on multiplying with indices.
To multiply for a given base, add the Positioned above and to the right of a number. It is an abbreviation of repeated multiplication, eg 7³ means 7 × 7 × 7. An index may be positive, negative, zero or fractional. Indices are also referred to as powers or exponents.:
𝑎\(^𝑚\) × 𝑎\(^𝑛\) = 𝑎\(^𝑚\)\(^+\)\(^𝑛\)
GCSE exam-style questions
- Simplify this expression:
𝑦⁸ × 𝑦⁷
𝑦¹⁵
The base of both terms is 𝑦.
When terms are multiplied with the same base, add the indices
(𝑎\(^𝑚\) × 𝑎\(^𝑛\) = 𝑎\(^𝑚\)\(^+\)\(^𝑛\)):
- 𝑦⁸ × 𝑦⁷ = 𝑦⁸⁺⁷
- 𝑦⁸ × 𝑦⁷ = 𝑦¹⁵
- Simplify this expression:
5𝑡⁴ × 2𝑡² × 3𝑡³
30𝑡⁹
Multiply the coefficients 5, 2 and 3:
- 5 × 2 × 3 = 30
Multiply 𝑡⁴, 𝑡² and 𝑡³ by adding the indices:
- 𝑡⁴⁺²⁺³ = 𝑡⁹
The expression 5𝑡⁴ × 2𝑡² × 3𝑡³ simplifies to 30𝑡⁹.
How to divide indices
The Agreed rules for simplifying expressions (involving multiplication, division and raising to a power) using index notation with the same base. are used when the The number or variable that is raised to a power when using index notation. is the same.
Follow the examples and working out below on dividing with indices.
To divide for a given base, subtract the Positioned above and to the right of a number. It is an abbreviation of repeated multiplication, eg 7³ means 7 × 7 × 7. An index may be positive, negative, zero or fractional. Indices are also referred to as powers or exponents.:
𝑎\(^𝑚\) ÷ 𝑎\(^𝑛\) = 𝑎\(^𝑚\)\(^–\)\(^𝑛\)
GCSE exam-style questions
- Find the value of 𝑛 when
𝑥\(^𝑛\) ÷ 𝑥⁵ = 𝑥².
𝑛 = 7
The base is the same (𝑥). Calculated the division by subtracting the indices:
- 𝑥\(^𝑛\) ÷ 𝑥⁵ = 𝑥\(^𝑛\)\(^–\)⁵
- 𝑛 – 5 = 2
Add 2 and 5 to find the value of 𝑛:
- 2 + 5 = 7
- 𝑛 = 7
- Simplify the expression 𝑞²⁰ ÷ 𝑞⁴.
𝑞²⁰ ÷ 𝑞⁴ = 𝑞¹⁶
The base is the same (𝑞).
Divide by subtracting the indices
(𝑎\(^𝑚\) ÷ 𝑎\(^𝑛\) = 𝑎\(^𝑚\)\(^–\)\(^𝑛\)):- 𝑞²⁰ ÷ 𝑞⁴ = 𝑞²⁰⁻⁴
- 𝑞²⁰ ÷ 𝑞⁴ = 𝑞¹⁶
How to raise a power to a power
The Agreed rules for simplifying expressions (involving multiplication, division and raising to a power) using index notation with the same base. are used when the The number or variable that is raised to a power when using index notation. is the same.
Follow the examples and working out below on raising a power to a power.
To raise to a A number that is multiplied by itself one or more times is raised to a power. 𝑛⁴ is ‘𝑛 to the power of 4’ and means 𝑛 × 𝑛 × 𝑛 × 𝑛. The power is the value of the index or exponent. for a given base, multiply the Positioned above and to the right of a number. It is an abbreviation of repeated multiplication, eg 7³ means 7 × 7 × 7. An index may be positive, negative, zero or fractional. Indices are also referred to as powers or exponents.:
(𝑎\(^𝑚\))\(^𝑛\) = 𝑎\(^𝑚\)\(^𝑛\)
GCSE exam-style questions
- Simplify and evaluate:
(2⁴)²
2⁸ = 256
To raise an index to a power, multiply the indices:
- 4 × 2 = 8
- (2⁴)² = 2⁸
Multiply eight 2s together:
- 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 256
- Simplify this expression:
(2𝑡⁴)³
8𝑡¹²
Everything in the bracket is cubed (raised to the power of three):
- 2 is cubed and 𝑡⁴ is cubed.
- 2³ is 2 × 2 × 2 = 8.
To raise an index to a power, multiply the indices:
- 4 × 3 = 12
- (𝑡⁴)³ = 𝑡¹²
(2t⁴)³ simplifies to 8𝑡¹².
Now you've revised laws of indices, why not look at order of operations and using negative numbers?
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