4p5 is a short way of writing 4 × p × p × p × p × p. The 4 is called the coefficient , the p is called thebase or variable and the 5 is called the index, power or exponent.
Indices (plural of index) obey certain rules when they are being multiplied and divided.
Rules of Indices |
Rules of Indices
Index of one
A number or variable to an index of 1, is equal to itself.
e.g. 41 = 4
y1 = y
Multiplication
When multiplying numbers or variables with indices, add the indices together.
The bases must be the same number or variable.
e.g a3 × a4 = a 3 + 4 = a7because a3 × a4 = (a × a × a) × (a × a × a × a) = a7
Division
When dividing numbers or variables with indices, subtract the indices. The bases must be the same number or variable.
Two indices
When raising a number or variable with an index to another index, multiply the two indices together.
e.g. (a3)2 = a3x2 = a6because (a3 )2 = (a × a × a) × (a × a × a) = a6
Index of zero
Any number or variable to the index of 0 is equal to 1. Note: 00 is undefined. i.e. it does not exist.
Negative index
A number or variable raised to a negative index is equal to the reciprocal of the number or variable to the same positive index.
Summary
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Multiplying (add the indices) |
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Dividing (subtract the indices) |
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Powers (multiply the indices) |
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Power of one (stays same) |
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Power of zero (always equals one) |
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Negative index (take the reciprocal) |
Powers of Ten
With a number multiplied by a power of 10, the index indicates the direction and the number of places the point must be moved.
e.g. 1.30 × 102 = 1.30 × 100 = 130 (point moves 2 places right)
0.0562 × 103 = 0.0562 × 1000 = 56.2 (point moves 3 places right)
7.64 × 10 -2 = 7.64 ÷ 100 = 0.0764 (point moves 2 places left)
Expressions involving Indices
To simplify algebraic expressions involving indices, use the rules of indices.
Simplify the signs first, numbers next and then the similar variables.
