INDICES
An index (power) is the small floating number that appears after a number or letter. It indicates a number has been multiplied by itself multiple times.
For instance
• 3 x 3 x 3 x 3 x 3 (3 multiplied by itself)
= 3⁵ (multiplied itself 5 times)
• 8 x 8 x 8 x 8
= 8⁴
• a x a x a
= a³
NB: An index can be any number, be it fraction or counting number
When working with indexed numbers, certain operations can be carried out. i.e you can multiply as well as divide. All these follow a certain principle called LAWS OF INDICES
LAWS OF INDICES
NB: In multiplication, when the base numbers are the same, just add the index numbers together. Same goes to division, instead of adding, you subtract.
But a° = 1 i.e any number with the power 0 is 1
E.g 2⁰ is 1; -3⁰ is also 1
Examples on multiplication of index form
First of, let's do a quick recap.
We know 3 x 3 x 3 x 3 x 3 x 3 = 3⁶ and vice versa
(a) 2³ x 2⁵ (b) (3³ x 5) x (4² x 3²) (c) 3° x 2³
Solution
(a) 2³ x 2⁵
Here we are multiplying; remember the law
a² x a³
= 2²+³
= 2⁵
(b) (3³ x 5) x (4² x 3²)
First thing here is to open the brackets
3³ x 5 x 4² x 3²
It's advisable to collect like terms here and rearrange.
= 3³ x 3² x 4² x 5
= 3³+² x 4² x 5
= 3⁵ x 4² x 5
NB: 3, 4 and 5 are of different bases. There's nothing left to do. That's our final answer.
(c) 3° x 2³
Remember our law a° = 1
Therefore 3° x 2³
= 1 x 2³
= 2³
Examples on division of index form
We already know the law
Evaluate each of these
(a) 4⁴ ÷ 4² (b) 5⁸ ÷ (5² x 5³) (c) 3⁹ ÷ 3²
(d) 3⁷ ÷9
Solution
(a) 4⁴ ÷ 4²
= 4⁴-²
= 4²
(b) 5⁸ ÷ (5² x 5³)
First things first, open the bracket
(5² x 5³)
This is multiplication, so we add the indexes.
5² x 5³
= 5²+³
= 5⁵
Recall the question is 5⁸ ÷ (5² x 5³)
Therefore 5⁸ ÷ 5⁵
= 5⁸-⁵
= 5³
(c) 3⁹ ÷ 3²
= 3⁹-²
= 3⁷
(d) 3⁷ ÷ 9
Okay, this is interesting. You might be wondering how we can solve this using the laws of indices because the two terms are of different bases. The best thing to do is cover the whole number (9) to index form. Best way to do that is to travel down the memory lane.
We find the factors of 9
9 in index form is 3²
Therefore 3⁷ ÷ 9 will be 3⁷ ÷ 3²
= 3⁷-²
= 3⁵
That's all for now, please comment and share your thoughts. Any question will surely be answered.
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Indices