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Indices, Powers and Roots

In $3^{4}$ , the raised 4 is the index. It means multiply four 3s together: $3 \times 3 \times 3 \times 3 = 81$ .Taking the fourth root reverses this: $481 = 3$ .

Year 7–10 (UK)Grade 6–9 (US)

By Lexie LiCo-founder of Maths Angel

You send a clip to 3 friends on Monday. Each person forwards it to 3 new people the next day. The clip reaches 9 new people on Tuesday, 27 on Wednesday and $3^{4} = 81$ on Thursday.

In $3^{4}$ , 3 is the base and 4 is the index. Read it as “3 to the power of 4”. This is called index notation. The plural of index is indices. An index is also called an exponent. Learn more: powers and indices, from square numbers to BIDMAS.

How to solve

Use these three groups of index laws to simplify powers with numbers or letters.

Multiplying and dividing powers
Same-base multiplication: add the indices. $2^{3} \times 2^{4} = 2^{3 + 4} = 2^{7} =$ $128$
Same-base division: subtract the indices. $4^{5} \div 4^{3} = 4^{5 - 3} = 4^{2} =$ $16$
Matching indices, different bases? Combine the bases: $2^{3} \times 5^{3} = (2 \times 5)^{3} = 1 0^{3} =$ $1, 000$
Learn more: multiplying and dividing powers with the same base or index.
Negative indices and the power of a power
Each time you subtract 1 from the index, divide the value by the base. Here, the base is 3.
A negative index means take the reciprocal. To calculate $3^{- 2}$ , divide 1 by $3^{2}$ : $3^{- 2} = \frac{1}{3 ^{2}} = \frac{1}{9}$
For a power of a power, multiply the indices: $(2^{3})^{5} = 2^{3 \times 5} = 2^{15}$
This also works with negative indices: $(3^{2})^{- 2} = 3^{2 \times (- 2)} = 3^{- 4} = \frac{1}{81}$
Learn more: negative indices and the power of a power.
When the index is a fraction: root first, then power
The denominator (bottom number) gives the root. The numerator (top number) gives the power. Taking the root first keeps the numbers small.
Example 1 $12 5^{\frac{2}{3}} = (3125)^{2} = 5^{2} = 25$
Example 2 $1 6^{\frac{3}{4}} = (416)^{3} = 2^{3} = 8$
Learn more: the nth root and fractional indices.

Common mistakes

The mistake	What to do instead
You worked out $2^{3}$ as $2 \times 3 = 6$ .	The index counts how many copies of the base are multiplied together: $2^{3} = 2 \times 2 \times 2 = 8$ .Never multiply the base by the index.
You wrote $5^{0} = 0$ .	Any number except 0 to the power of 0 equals 1, so $5^{0} = 1$ and $1 0^{0} = 1$ .
You simplified $a^{3} \times a^{2}$ to $a^{6}$ .	Same-base multiplication adds the indices: $a^{3} \times a^{2} = a^{3 + 2} = a^{5}$ . Multiplying indices is reserved for a power of a power: $(a^{3})^{2} = a^{6}$ .
You gave $2^{- 3}$ as a negative number.	A negative index means take the reciprocal: $2^{- 3} = \frac{1}{2 ^{3}} = \frac{1}{8}$ . Here, the answer is positive.
You halved 144 and wrote $144 = 72$ .	Square-rooting asks which positive number multiplied by itself gives 144: $144 = 12$ , because $12 \times 12 = 144$ . Halving is a different operation. The definition is settled in basics of square roots.
You simplified $72$ to $218$ and stopped.	Pull out the largest square factor: $72 = 36 \times 2 = 62$ . If a square factor still hides under the root, keep going. The method is in multiplying and dividing square roots.

FAQs

Are there 6, 7 or 9 laws of indices?

Different websites group the same rules differently. Here are seven common rules:

$a^{m} \times a^{n} a^{m} \div a^{n} a^{n} \times b^{n} a^{n} \div b^{n} (a^{m})^{n} a^{- n} a^{\frac{1}{n}} = a^{m + n} = a^{m - n} = (ab)^{n} = (a \div b)^{n} = a^{mn} = \frac{1}{a ^{n}} = n a$

You cannot divide by zero, and a base with a negative index cannot be zero. Two more rules are $a^{1} = a$ and $a^{0} = 1$ , with $a \neq = 0$ for the second rule.

How do you know which law of indices to use?

Look at the bases, indices and operation.

When multiplying powers with the same base, add the indices; when dividing, subtract them.
When powers have the same index, multiply or divide the bases and keep the index.
For a negative index, work out the positive power, then divide 1 by the result.
In a fractional index, the bottom number gives the root and the top number gives the power.

If several rules apply, use one at a time.

Why is any non-zero number to the power of 0 equal to 1?

The division law forces it. Work out $2^{3} \div 2^{3}$ both ways: subtracting the indices gives $2^{0}$ , and dividing the actual numbers gives $8 \div 8 = 1$ . Both answers describe the same calculation, so $2^{0}$ must equal 1, and the argument repeats for any non-zero base. The value 1 is the only one that keeps the laws consistent.

How are powers and roots connected?

Squaring 5 gives 25, and the square root of 25 is 5. Cubing 2 gives 8, and the cube root of 8 is 2. Fractional indices express roots: a power of $\frac{1}{2}$ means a square root, and a power of $\frac{1}{3}$ means a cube root.

What are indices actually used for?

Indices appear in area and volume units, such as m² and cm³. Computer security uses powers of 2: a 128-bit key has $2^{128}$ possible values. Scientists use standard form to write very large or small numbers. For example, the Moon is about 384,000 km from Earth: $3.84 \times 1 0^{5}$ km. Learn how standard form works.

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How to solve

Multiplying and dividing powers

Negative indices and the power of a power

When the index is a fraction: root first, then power

Key skills

Common mistakes

FAQs

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