05Aindexlaws

= Basic Index Laws =

Recall that index form looks like this:


 * The ** index ** is also called the ** power ** or the ** exponent **
 * The ** base ** is the main number which has been raised to a power


 * 1st Index Law **

... When __**multiplying**__ numbers in index form **__with the same base__**:
 * keep the base and __**ADD**__ the indices

This is because when mutliplying, we can expand the numbers into factor form:


 * Example 1 **

... Simplify each of the following:

... ** 1) ** . a 5 × a 4 ... ... ... = a 9

... ** 2) ** . 5b 6 × 2b 2 ... ... ... = 10b 8

... ** 3) ** . a 4 b 8 × a 4 b ... ... ... = a 8 b 9

... When __**dividing**__ numbers in index form **__with the same base__**:
 * 2nd Index Law **
 * keep the base and __**SUBTRACT**__ the indices

This is because, when dividing, we can expand the numbers into factor form and then cancel:


 * Example 2 **

... Simplify each of the following:

... ... ** 1) ** . a 7 ÷ a 3 . math . \qquad \quad = \dfrac{a^7}{a^3} \\ . \\ . \qquad \quad = a^4 math

... ... ** 2) ** . 10a 5 ÷ 2a 4 . math . \qquad \quad = \dfrac{10a^5}{2a^4} \\ . \\ . \qquad \quad = 5a^1 \\ . \\ . \qquad \quad = 5a math

... ... ** 3) ** . a 9 b 6 ÷ a 3 b . math . \qquad \quad = \dfrac{a^9b^6}{a^3b} \\ . \\ . \qquad \quad = a^6b^5 math

... ... ** 4) ** . 8a 7 b 3 ÷ 6a 2 b 8 . math . \qquad \quad = \dfrac{8a^7b^3}{6a^2b^8} \\ . \\ . \qquad \quad = \dfrac{4a^5}{3b^5} math


 * 3rd Index Law (Zero Index) **

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

This is because of the pattern in these two examples The same pattern will occur with any base.


 * Example 3 **

... Simplify the following:

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

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

... ** 3) ** .. (5x) 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

This is because: 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 12.

... Simplify the following
 * Example 4 **

... ... ** 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 Index Law (Brackets) **

... When __**raising**__ __**one**__ term in **__brackets__** to a power:
 * every number inside the bracket is raised to that power
 * (ab) 3 = a 3 b 3


 * 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 \quad = \dfrac{3^3 \times a^{15}}{b^3} \\ . \\ . \qquad \quad = \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

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