Number
Grade 4-6Indices & Roots
Indices questions are built on a small set of rules, and once those rules are learned they apply to every base, whether it is a number or a letter. This lesson covers multiplying and dividing powers, raising a power to a power, zero and negative indices, and roots including fractional indices.
What you need to know
- Multiplying powers of the same base adds the indices; dividing subtracts them.
- A power raised to a power multiplies the indices.
- Anything to the power 0 is 1; a negative power means one over the positive power.
- A fractional power is a root: x to the power one half is root x, x to the power two thirds is the cube root of x squared.
Written by Asad, co-founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
Multiplying and dividing powers
When multiplying powers of the same base, add the indices. When dividing powers of the same base, subtract the indices. This only works when the base is the same on both sides.
3⁴ × 3² = 3⁴⁺² = 3⁶. 5⁷ ÷ 5³ = 5⁷⁻³ = 5⁴.
Raising a power to a power
When a power is raised to another power, multiply the two indices together.
(2³)⁴ = 2³ˣ⁴ = 2¹².
Zero and negative indices
Any non-zero number raised to the power 0 is equal to 1. A negative index means finding the reciprocal of the number raised to the positive version of that index.
7⁰ = 1. 2⁻³ means 1 ÷ 2³, which is 1/8.
Roots and fractional indices
A square root undoes squaring, and a cube root undoes cubing. A fractional index of 1/n means the nth root of the number. A fractional index of m/n means finding the nth root first, then raising the result to the power m.
√144 = 12, since 12² = 144. 25^(1/2) = 5, since this is the square root of 25. 27^(2/3): find the cube root of 27 first, which is 3, then square it, giving 9.
Worked examples
Three exam-style questions, fully solved
Simplify 3⁴ × 3². Give your answer as a single power of 3.
Easy- 1.Since the base is the same on both sides, add the indices: 4 + 2
Answer: 3⁶
Work out 2⁻³. Give your answer as a fraction.
Medium- 1.A negative index means finding the reciprocal of the positive power: 1 ÷ 2³
- 2.Work out 2³ = 8
Answer: 1/8
Work out 27^(2/3).
Hard- 1.The denominator of the fractional index tells you which root to find: the cube root of 27
- 2.Find the cube root of 27: 27^(1/3) = 3
- 3.The numerator tells you the power to raise the result to: 3² = 9
Answer: 9
Practice
6 questions, marked instantly
Type an answer and check it. The worked solution appears once you have had a go. No login needed, and your progress saves in this browser.
Practice
Now try these yourself.
Type your answer and check it. The worked solution appears once you have had a go.
Work out 2⁵.
Simplify x³ × x⁴. What is the power of x in the answer?
Work out (2³)².
Work out 5⁰.
Work out the cube root of 64.
Write 3⁻² as a fraction.
Avoid these
The mistakes students make most often
Adding or subtracting indices when the bases are different, which only works when the base is exactly the same on both sides.
Writing any number to the power 0 as 0, instead of the correct value of 1.
Treating a negative index as making the answer negative, instead of taking the reciprocal of the positive power.
When raising a power to a power, adding the indices instead of multiplying them, for example writing (2³)⁴ as 2⁷ instead of 2¹².
With a fractional index like 27^(2/3), only applying the root or only applying the power, instead of doing both steps.
FAQ
Questions students and parents ask
Before this topic, make sure you know
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