What are the law of Indices?
A power, or an index, is used to write a product of numbers very compactly. The plural of index is indices.
Simplifying Indices
Index Laws 1 - multiplying
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Example 1:
Simplify 4^5 x 4³. (The ^ symbol = 'the power of')
You can simplify by doing: 4 x 4 x 4 x 4 x 4 x 4 x 4 x 4 = 4^8
Alternatively, you may notice that 5 + 3 = 8, so when multiplying terms with the same base you can add the powers! The video to the right shows you a step by step tutorial. |
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Index Laws 2 - dividing
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Example 2:
Simplify 9^5 ÷ 9³ (The ^ symbol = 'the power of')
To simplify with division you will still need to multiply but instead of multiplying the factors in the next step, you'll need to divide.
e.g. 9 x 9 x 9 x 9 x 9 ÷ 9 x 9 x 9 = 9² You may notice that the powers 5 and 3, when 3 is subtracted by 5, you end up with 2. When dividing terms with the same base, you subtract the powers. The video to the right gives you a step by step tutorial. |
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Index Laws 3 - raising powers
Example 3:
Simplify (4^3)^4 (The ^ symbol = 'the power of')
To simplify with raising powers you need to multiply the powers e.g. 4 x 4 x 4 x 4 x 4 x 4 x 4 x 4 x 4 x 4 x 4 x 4 = 4^12.
You may notice that there are four sets of three and you can also simplify by doing 4³ x ^4 = 4^12. Video coming soon! |
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Index Laws - with algebra (letters)
Example 4:
Simplify 5a³ x 6a^7 (The ^ symbol = 'the power of')
To simplify these type of index, you will need to multiply the number by the letter depending on the power. For example, 5 x a x a x a x 6 x a x a x a x a x a x a x a. The sum will then look like 5 x 6 x a³+^7 which would give you 5 x 6 which is 30 + a = 30a and 3 + 7 = 10 so 30a^10.
You may notice that you can times together the numbers, as a is on both sides it turns into one a and + 7 and 3 together to make 30a^10. |
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Indices with Algebraic Expressions
You may need to use some paper to help with the working out in these three videos. Feel free to go through the video alongside Mr Hegarty to understand fully.
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