An index, power or exponent is a way of shortening expressions such as 3 × 3 × 3 × 3 × 3 × 3 × 3 to 37
4p5 is a short way of writing 4 × p × p × p × p × p.
The 4 is called the coefficient , the p is called the base or variable and the 5 is called theindex, power or exponent.
Index Facts
The value of a variable to the power of 0 is always 1.
e.g. x0 = 1
A variable to the power of 1 is the variable itself.
e.g. x1 = x
Indices (plural of index) obey certain rules when they are being multiplied and divided.
Multiplication
When multiplying numbers or variables with indices, add the indices together.
The bases must be the same number or variable.
When multiplying add the indices. a3 × a4 = a 3 + 4 = a7 Coefficients are multiplied as normal. 3p3 × 4p2 = 12p5
Division
When dividing numbers or variables with indices, subtract the indices.
The bases must be the same number or variable.
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Method
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Examples
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| When dividing subtract the indices. | p5⁄p3 = p5−3 = p2 |
| Coefficients are divided as normal. | 12x6⁄4x2 = 3x6−2 = 3x4 |
Powers in Brackets
When a term inside a bracket is raised to a power, for example (x2)3 then the two powers aremultiplied .
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Method
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Examples
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| For a power to another power multiply the indices. | (p2)3 = p2×3 = p6 |
| Coefficients are calculated as normal exponents. | (2q3)4 = 16q12 |
Roots
When the root of a variable to an exponent is taken, the exponent is divided. i.e. For the squareroot, divide the exponent by 2.
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Method
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Examples
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| For the root of an exponent divide the index. | √p6 = p6÷2 = p3 |
| Coefficients are calculated as normal roots. |
√(16p8) = 4p8÷2 = 4p4 |
