Indices are variables raised to a power. The laws of indices show how to manipulate expressions involving powers, including those which have negative or fractional powers.
Negative indices
When an index has a negative power, this means that it is one divided by the index raised to the positive of the power. This is also called the reciprocal of the index. You can use this rule:
a−n=an1
Example 1
What is 4−2?
4−2=421=161
Fractional indices
Fractional indices are indices whose powers are expressed in fractions. The denominator of the fraction refers to the root of the index. The answer is then raised to the power, which is the value of the numerator.
anm=(na)m
Common indices
Some of the most commonly used indices are as follows:
Reciprocal of a
a1
a−1
Square root of a
2a
a21
Cube root of a
3a
a31
Example 2
What is (4936)21?
(4936)21=4936=76
Example 3
What is 16−23?
16−23=16231=1621×31=(16)31=431=641
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Writing indices and index laws
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Index laws: negative and fractional indices - Higher
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FAQs - Frequently Asked Questions
How do you write a square root in index form?
To write a square root in index form, raise the index to the power of a half.
What are fractional indices?
Fractional indices have powers that are fractions. The denominator of the fraction refers to the root of the index. The answer is then raised to the power, which is the value of the numerator.
What are negative indices?
Negative indices are variables raised to a negative power. This can be written as one divided by, or the reciprocal of, the index.