01Aindex-3-4

= Basic Index Laws (continued) =


 * Recall: **




 * 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

Zero Index
Notice the pattern in these two examples:
 * continuing the pattern in each case results in
 * 10 0 = 1
 * 2 0 = 1
 * This pattern will be repeated for __**any**__ base


 * 3rd Index Law (Zero Index) **

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


 * Example 1 **

... Simplify the following:

... ** 1) ** .. x 0 ... ... ... = 1

... ** 2) ** .. 5x 0 ... ... ... = 5 × 1 ... ... {due to BODMAS}  ... ... .... = 5

... ** 3) ** .. (5x) 0 ... ... ... .. = 1


 * Notes: **
 * In ** (2) **, BODMAS says that "** powers of **" is more important than "** multiply **"
 * so calculate x 0 first, then multiply by 5

** Raising a Number in Index Form to Another Power **
In the same way that
 * a 3 = a × a × a

We can write (m 4 ) 3 as
 * (m 4 ) 3 = m 4 × m 4 × m 4 ... ... {now use 1st Index Law}
 * ... ... .. = m 4+4+4
 * ... ... .. = m 12.

The result is that we have multiplied the indexes together
 * 4 × 3 = 12


 * 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

... Simplify the following
 * Example 2 **

... ... ** 1) ** . (b 2 ) 4 ... ... ... ... . = b 8

... ... ** 2) ** . (c 3 ) 2 × (c 4 ) 4 ... ... ... ... . = c 6 × c 16 ... ... {now use 1st index law}  ... ... ... ... .. = c 22


 * 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

Some people consider the bracket involving division to be a different law (6th index law)


 * Example 2 **

... Simplify the following

... ... ** 1) ** . (a 2 b 3 ) 4 ... ... ... ... . = (a 2 ) 4 × (b 3 ) 4 ... ... {now use 4th index law}  ... ... ... ... . = a 8 b 12

... ... ** 2) ** . (5a 3 ) 2 ... ... ... ... . = 5 2 a 6  ... ... ... ... .. = 25a 6

... ... ** 3) ** math \\ . \qquad \quad \left( \dfrac{3a^5}{b} \right)^3 \\ . \\ . \qquad \qquad = \dfrac{3^3 \times a^{15}}{b^3} \\ . \\ . \qquad \qquad = \dfrac{27a^{15}}{b^3} math

... This law does __**not**__ apply when there are two (or more) terms in a bracket (seperated by + or –)
 * Note: **
 * (a + b) 2 ** ≠ ** a 2 + b 2


 * Example 3 **

... Simplify the following:

... ... ** 1) ** . (2a 4 b) 3 × (3ab 2 ) 2 ... ... ... .. = 8a 12 b 3 × 9a 2 b 4  ... ... ... .. = 72a 14 b 7.

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