IGCSE Mathematics: Index Laws Practice Questions
When multiplying powers of the same base you add the indices, when dividing you subtract them, and when raising a power to a power you multiply them. A negative index means a reciprocal and a fractional index means a root.
Topic 2.4 of Cambridge IGCSE Mathematics 0580 appears in short recall questions and inside longer algebra. Fractional indices are the part that costs marks, because both the root and the power must be applied. The questions below cover the laws and both special cases.
What you need to know for Indices
- Multiplying and dividingxa times xb = xa+b, and xa divided by xb = xa-b. The base must be the same for either law to apply.
- Power of a power(xa)b = xab. Any number in front of the bracket is raised to the outer power as well.
- Zero indexAny non-zero number or expression raised to the power 0 equals 1.
- Negative indexx-n equals 1 divided by xn. The answer is a fraction, not a negative number.
- Fractional indexThe denominator gives the root and the numerator gives the power. 272/3 means the cube root of 27, then squared.
- Order of operationsTake the root first when evaluating a fractional index. The numbers stay much smaller that way.
IGCSE Mathematics Indices questions and answers
4 exam-style questions written to the 0580 syllabus. Try each one on paper first, then open the worked answer to check your method against the marks.
Simplify (3x2y3) multiplied by (4x5y).
Show the worked answer
- Multiply the numerical coefficients: 3 times 4 = 12.
- For the x terms, add the indices: x2 times x5 = x7.
- For the y terms, add the indices, remembering that y means y1: y3 times y1 = y4.
- The simplified expression is 12x7y4.
Simplify (2x3)4.
Show the worked answer
- Everything inside the bracket is raised to the power 4, including the coefficient.
- The coefficient: 24 = 16.
- The variable: (x3)4 means multiply the indices, so x12.
- The simplified expression is 16x12.
Evaluate 272/3 without a calculator, showing your method.
Show the worked answer
- The denominator of the fractional index gives the root, so take the cube root first.
- The cube root of 27 is 3, because 3 x 3 x 3 = 27.
- The numerator gives the power, so square that result: 32 = 9.
- 272/3 = 9. Taking the root first keeps the arithmetic small. Squaring first would mean finding the cube root of 729.
Evaluate 16-1/2 without a calculator.
Show the worked answer
- The negative sign means take the reciprocal, so 16-1/2 = 1 divided by 161/2.
- The index one half means the square root.
- The square root of 16 is 4.
- So the answer is 1 divided by 4, which is one quarter. Note the answer is positive, because a negative index produces a fraction, not a negative value.
Common mistakes in this topic
- Multiplying indices when multiplying powers.
- Forgetting to raise the coefficient when a bracket has an outer index.
- Treating a negative index as a negative answer.
- Multiplying by a fractional index instead of taking a root.
- Applying index laws to terms with different bases.
Exam tips
- Deal with the coefficients and each letter separately, then recombine.
- For fractional indices, take the root first and the power second.
- Remember that a letter with no visible index has an index of 1.
- Check negative index answers are positive fractions.
- Write the index law you are applying beside your working. The rule itself often earns a method mark.
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Indices FAQs
What does a negative index mean?
A negative index means take the reciprocal of the positive power. So x to the power minus 3 equals 1 divided by x cubed, and 16 to the power minus one half equals one quarter. The result is a fraction, never a negative number.
What does a fractional index mean?
The denominator gives the root and the numerator gives the power. So 27 to the power two thirds means take the cube root of 27, which is 3, then square it to get 9. Taking the root first keeps the numbers manageable.
Why does anything to the power zero equal 1?
Dividing a power by itself gives 1, and the division law says the indices subtract to zero. So x cubed divided by x cubed is both 1 and x to the power zero, which means x to the power zero must equal 1 for any non-zero x.
Do index laws work when the bases are different?
No. The laws for adding and subtracting indices only apply when the base is identical. An expression such as 2 cubed times 3 squared cannot be simplified using index laws, so it must be evaluated as 8 times 9, which is 72.
This page covers the index laws applied to algebra. For squares, cubes and roots as numerical values, see Powers and Roots.
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Written to the published Cambridge IGCSE Mathematics (0580) syllabus. Check your school entry code and syllabus year, because Core and Extended candidates are assessed on different content. Last reviewed 2026-08-12.