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Indices - Revision Notes

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7.1Indices


In these revision notes for Indices, we cover the following key points:

  • What are indices? What is the other name used for indices?
  • How indices are related to powers?
  • What are the properties of indices?
  • What is the reciprocal of a number?
  • How is the reciprocal involved in raising a number in a negative power?
  • How to calculate the powers of negative numbers?
  • What differs when raising a negative number to an odd power to raising the corresponding positive number at the same power?

Indices Revision Notes

If the same factor is multiplied several times by itself, we use the term "power" to represent such a recurring multiplication by the same number. Thus, the power of a number tells us how many times to use that number in a multiplication. The recurring factor is called the base, the number that shows how many times this factor appears in a recurring multiplication is called the exponent and the result of this operation is called power. Another name used for the exponent is index (indices in plural).

Math Tutorials: Indices Example

Indices (exponents) have the following properties:

Property 1: When multiplying powers with the same base, we add the indices.

In symbols,

am × an = am + n

Property 2: When dividing powers with the same base, we subtract the indices.

In symbols,

am ÷ an = am ÷ n

Property 3: When raising a power into another power, we multiply the indices without changing the base.

In symbols,

(am )n = am × n

Property 4: If two numbers of different bases are raised at the same power, the bases multiply without changing the index.

In symbols,

ac× bc = (a × b)c

Property 5: Any number raised to the first power gives the number itself.

In symbols,

a1 = a

Property 6: Any number raised at power zero gives 1.

In symbols,

a0 = 1

By definition, the reciprocal represents a number, expression, or function so related to another that their product is unity (one, therefore). In other words, the reciprocal of a quantity represents the quantity obtained by dividing the number one by a given quantity.

There is a property involving negative indices that says:

If a number or expression is raised to a negative power, it is equal to the reciprocal raised at the corresponding positive power.

In symbols,

a - b = (1/a)b

In fractions, the reciprocal is obtained by swapping the positions of numerator and denominator.

If a negative number is raised at an even power (the index is an even number), the result is positive and when a negative number is raised at an odd power (the index is an odd number) the result is negative.

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