01Dindexlaws

= Summary of Index Laws =


 * 1st Index Law (Multiplying) **

... When __**multiplying**__ numbers in index form **__with the same base__**:
 * keep the base and __**ADD**__ the indices
 * a 5 × a 3 = a 8


 * 2nd Index Law (Dividing) **

... When __**dividing**__ numbers in index form **__with the same base__**:
 * keep the base and __**SUBTRACT**__ the indices
 * a 7 ÷ a 3 = a 4


 * 3rd Index Law (Zero Index) **

... __**Anything**__ raised to the power of zero equals one
 * a 0 = 1


 * 4th Index Law (Raising to Another Power) **

... When __**raising**__ numbers in index form **__to another power__**:
 * keep the base and __**MULTIPLY**__ the indices
 * (a 5 ) 3 = a 15


 * 5th & 6th Index Law (Brackets) **

... When __**raising**__ __**one**__ term in **__brackets__** to a power: math . \quad \cdot \quad \left( \dfrac{a}{b} \right)^4 = \dfrac{a^4}{b^4} math
 * every number inside the bracket is raised to that power
 * (ab) 3 = a 3 b 3


 * 7th Index Law (Negative Indices) **

... A number with a __**negative**__ index is equal to the reciprocal of the same number with a **__positive__** index: math . \qquad \bullet \quad a^{-3} = \dfrac{1}{a^3} math


 * 8th Index Law (Fractional Indices) **

... When the __**index**__ of a is the fraction 1/n, the number is equivalent to the nth root of a: math \\ . \qquad \cdot \quad a^{\frac{1}{n}} = \sqrt[n]{a} \\. \\ . \qquad \cdot \quad a^{\frac{1}{3}} = \sqrt[3]{a} math

Combining this with the 4th index law {raising a number in index form to another power} we get: math . \qquad \qquad a^{\frac{m}{n}} = \Big( a^{\frac{1}{n}} \Big)^m = \Big( \sqrt[n]{a} \Big)^m math or math . \qquad \qquad a^{\frac{m}{n}} = \Big( a^m \Big)^{\frac{1}{n}} = \sqrt[n]{ a^m } math

= Combining Index Laws =

In most situations, simplifying indices involves using more than one of the index laws
 * Follow BODMAS {Brackets, Of (powers of), Multiply&Divide, Add&Subtract}
 * This means expand any brackets first
 * an index is more important than multiply so:
 * 3a 2 is not the same as (3a) 2.
 * If an expression is a fraction, simplify the numerator and the denominator seperately first and then divide:
 * simplify across and then down.
 * Don't forget that when dividing two fractions, flip the second fraction and multiply
 * Evaluate simple powers of integers (eg 2 3 = 8) but leave hard powers in index form (eg 7 8 )
 * The instruction to "Evaluate" means find the numerical answer (as an integer or a fraction)
 * The instruction to "Simplify" means your answer will probably be in index form
 * Write the final answer using positive indices

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