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Home > Maths II > Algebra > Indices

Maths II

Indices

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Every year, without fail, examiners report of too many errors by exam candidates in dealing with indices. This is in part due to weaknesses in working with fractions and negative numbers. But not knowing about the laws of indices and not practising how to apply them contributes to these errors.

Two basic laws of indices

  • Multiplication law:

    a to the power of m times a to the power of n equals a to the power of m plus n.

  • Division law:

    a to the power of m divided by a to the power of n equals m to the power of m minus n.

Using these laws - Example 1

Question

y to the power of 7 times y to the power of 3 times y to the power of 5

Answer

Use the multiplication law. This tells you to add the indices.

y to the power of seven times y to the power of 3 times y to the power of 5 equals y to the power of 7 plus 3 plus 5 equals y to the power of 15

Example 2

Question

y to the power of 10 divided by y to the power of 3

This could also be written as

y to the power of 10 over y to the power of 3

Answer

Use the division law. This tells you to subtract the indices.

y to the power of 10 divided by y to the power of 3 equals y to the power of 10 minus 3 = y to the power of 7

Writing this example out in full gives

y times y times y times y times y times y times y times y times y times y times y over y times y times y

 

equals y multiplied by y multiplied by y multiplied by y multiplied by y multiplied by y multiplied by y.

equals y to the power of 7.

Example 3

Question

y to the power of 7 multiplied by y to the power of 4 over y to the power of 5

Answer

Find the index of the numerator first. Using the multiplication law, this is 7 + 4.

Two students in an exam. The girl at the front looks fed up. The boy behind her looks happy and is punching the air.

y to the power of 7 multiplied by y to the power of 4 over y to the power of 5 equals y to the power of 7 plus 4 over y to the power of 5 equals y to the power of 11 over y to the power of 5.

You'll impress the marker in an exam if you write down this intermediate stage! Now use the division law and subtract the indices.

Y to the power of 11 over y to the power of 5 equals y to the power of 11 minus 5 equals y to the power of 6.

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