Indices is the plural of index. An index is the small number telling us how many times to multiply a number by itself (e.g. 23 means 2 x 2 x 2). Indices make calculations more manageable by simplifying expressions and helping to solve equations. One of the challenging aspects of this higher maths topic is negative and fractional indices.
This guide focuses on the key points to learn about negative and fractional indices: what they mean, how they work and how to simplify expressions through their use. It is aimed at GCSE students to support their GCSE Maths revision. It is suitable for all major exam boards, including AQA and Edexcel. Quiz questions and their answers are included to test your knowledge.
If you need further support, TeachTutti has GCSE Maths tutors who can help you revise this topic.
The Laws of Indices
Indices are also called powers or exponents. They are a shorthand for repeated multiplications. For example, rather than writing 3 x 3 x 3 x 3, you can write 34, which equals 81. They simplify expressions and let us work efficiently with large numbers.
Laws govern the use of indices. It is essential to follow the laws of indices, which include:
- Multiplication law: When multiplying terms that share the same base, add the powers together e.g. am × an = am+n.
- Division law: When dividing terms with the same base, subtract the powers e.g. am ÷ an = am-n.
- Power of a power law: When you need to raise a power to another power, multiply the indices together. For example, (am)n = am×n.
You need to understand these key laws as negative and fractional indices are an extension of these basic rules. For further reading, TeachTutti has written an article about the laws of indices.
1
What is the correct simplification of 34 × 32?
Negative Indices
Negative indices follow a simple rule: a negative power represents the reciprocal of the positive power. If we have a-n, this is equal to 1/an. Put in another way, the negative sign in the exponent flips the base to its reciprocal while keeping the power positive.
For example:
- 2-3 = 1/23 = 1/8
- x-2 = 1/x2
Using negative indices lets us rewrite expressions in a more manageable form. This includes simplifying complete expressions that have both positive and negative powers. For instance:
- 53 × 5-2 simplifies to 53-2 = 51 = 5
Always remember that the negative sign in the power doesn't affect the base. Instead, it tells us to take the reciprocal of the base raised to the positive power.
Negative indices also interact with the laws of indices. For instance:
- Division law: am × a-n = am-n
- Multiplication law: am / a-n = am+n
Negative indices are useful when using very small values. For example, 10-3 is 1/1000 = 0.001, which is far more complex.
2
What is the correct value of 5-2?
Fractional Indices
Fractional indices work by combining the combined roots and powers in a single expression. An example of fractional indices is am/n. This can be broken down into two parts:
- The denominator n is the root, such as the square root or cube root.
- The numerator m tells us the power we need to raise the result.
For example, am/n can be written as (a1/n)m. You can calculate the root first (using 1/n) or apply the power first (m). It doesn't matter the order you choose as both lead to the same result.
For example:
- 161/ 2 = 4 (square root of 16).
- 271/3 = 3 (cube root of 27).
- 82/3 = (81/3)2 = 22 = 4.
Fractional indices are helpful when we use roots and powers in algebra because they simplify complex expressions. For example, rather than writing "the square root of x3", you can express it as x3/2.
Make sure you don't apply the numerator and denominator in the wrong order. For instance, in 163/4, students will sometimes accidentally cube 16 first and then take the fourth root. Always apply the root first with the denominator and then raise the result to the power with the numerator.
3
Simplify 641/3.
Combining Negative and Fractional Indices
You can sometimes expect to encounter an expression containing both fractional and negative indices. You can break down these complicated expressions into a few manageable steps:
- Start with the negative sign - Find the reciprocal bit first.
- Tackle the fractional part - Apply the root and power as required.
For example, break down 16-3/4:
- The negative sign turns it into 1/163/4.
- The fractional index means we need to find the fourth root of 16 and cube it. 16
- 3/4
- This means 16-3/4 can be written simply as 1/8.
Another example is 27-2/3:
- The negative sign turns it into 1/272/3.
- The fractional power means we must find the cube root of 27 and then square it. 272/3 = (the cube root of 27)2 = 32 = 9.
- Therefore, 27-2/3 can be written as 1/9.
Always use a methodical process such as this. It avoids mistakes and helps you follow your workings when tackling exam questions.
4
Simplify 8-2/3.
Practice questions 1 - Negative Indices
Now that we have explored negative and fractional indices, it is time to practice. We have given you three questions below on negative indices to test your understanding.
5
Simplify 10-2.
6
Simplify y-3 × y5.
7
Simplify 43 / 4-2.
Practice questions 2 - Fractional Indices
Try the three questions below on fractional indices. You can refer to the glossary if you don't understand any terms.
8
Simplify 91/2.
9
Simplify 642/3.
10
Simplify 274/3.
- First, we need to find the cube root of 27 and then raise it to the power of 4: 274/3 = (the cube root of 27)4 = 34 = 81.
Final thoughts
Negative and fractional indices are the most challenging index laws. However, a step-by-step approach coupled with practice makes these problems far more approachable. If you need further practice, MathsGenie has past paper questions on indices to test your knowledge. You can also read the BBC Bitesize article on fractions if you need to refresh your understanding of this topic.
If you need 1-2-1 support with this topic, such as how a fraction is simplified with indices, TeachTutti has GCSE Maths tutors to aid your revision. Lessons can be online or in-person. All tutors have an enhanced DBS check and are qualified in the subjects they teach.
Glossary
- Indices - How many times a number (called the base) is multiplied by itself e.g. 23 means 2 x 2 x 2.
- Bases - The number that is being raised to a power. For example, in 32, the base is 3.
- Exponent - The number that says how many times to multiply the base by itself e.g. the exponent is 2 in 32.
- Negative indices - This shows the reciprocal of positive powers. For example, x-2 equals 1 divided by x2.
- Fractional indices - Combine roots and powers in one expression e.g. a to the power of 1/2 means the square root of a. Meanwhile, a to the power of 2/3 means the cube root of a squared.
- Root - The number that produces the original whole number when multiplied by itself e.g. the square root of 16 is 4 (4 x 4).
- Cube root - A specific root that equals the original number when cubed. For instance, the cube root of 27 is 3 (3 x 3 x 3).
- Square root - A root that equals the original number when squared e.g. the square root of 9 is 3 (3 x 3).
- Numerator - The top number in a fraction. This represents the power in fractional indices, the numerator represents the power. For instance, in a2/3, 2 is the numerator.
- Denominator - The bottom number in a fraction. This is the root in fractional indices e.g. in a2/3, the denominator tells us to cube a.
- Laws of indices - Rules that tell us how to simplify expressions using powers. They include multiplication, division and raising powers to other powers.
- Reciprocal - The opposite of a number or term. For example, the reciprocal of x is 1 / x and the reciprocal of x2 is 1 / x.
