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Laws of Indices (DP IB Analysis & Approaches (AA)): Revision Note

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Laws of Indices

What are the laws of indices?

  • Index laws are rules for doing operations with powers

    • They work on both numbers and algebra

Law

Description

How it works

a to the power of 1 equals a

Anything to the power of 1 is itself

6 to the power of 1 equals 6

a to the power of 0 equals 1

Anything to the power of 0 is 1

8 to the power of 0 equals 1

a to the power of m cross times a to the power of n equals a to the power of m plus n end exponent

To multiply indices with the same base, add their powers

4 cubed cross times 4 squared equals open parentheses 4 cross times 4 cross times 4 close parentheses cross times open parentheses 4 cross times 4 close parentheses equals 4 to the power of 5

a to the power of m divided by a to the power of n equals a to the power of m over a to the power of n equals a to the power of m minus n end exponent

To divide indices with the same base, subtract their powers

7 to the power of 5 divided by 7 squared equals fraction numerator 7 cross times 7 cross times 7 cross times up diagonal strike 7 cross times up diagonal strike 7 over denominator up diagonal strike 7 cross times up diagonal strike 7 end fraction equals 7 to the power of 3 space end exponent

open parentheses a to the power of m close parentheses to the power of n equals a to the power of m n end exponent

To raise indices to a new power, multiply their powers

open parentheses 14 cubed close parentheses squared equals open parentheses 14 cross times 14 cross times 14 close parentheses cross times open parentheses 14 cross times 14 cross times 14 close parentheses equals 14 to the power of 6

open parentheses a b close parentheses to the power of n equals a to the power of n b to the power of n

To raise a product to a power, apply the power to both numbers, and multiply

open parentheses 3 cross times 4 close parentheses squared equals 3 squared cross times 4 squared

open parentheses a over b close parentheses to the power of n equals a to the power of n over b to the power of n

To raise a fraction to a power, apply the power to both the numerator and denominator

open parentheses 3 over 4 close parentheses squared equals 3 squared over 4 squared equals 9 over 16

a to the power of negative 1 end exponent equals 1 over a

a to the power of negative n end exponent equals 1 over a to the power of n

A negative power is the reciprocal

6 to the power of negative 1 end exponent equals 1 over 6

11 to the power of negative 3 end exponent equals 1 over 11 cubed

open parentheses a over b close parentheses to the power of negative n end exponent equals open parentheses b over a close parentheses to the power of n equals b to the power of n over a to the power of n

A fraction to a negative power, is the reciprocal of the fraction, to the positive power

open parentheses 2 over 5 close parentheses to the power of negative 3 end exponent equals open parentheses 5 over 2 close parentheses cubed equals 5 cubed over 2 cubed equals 125 over 8

a to the power of 1 over n end exponent equals n-th root of a

The fractional power 1 over n is the nth root ( n-th root of blank)

25 to the power of 1 half end exponent equals square root of 25 equals 5

27 to the power of 1 third end exponent equals cube root of 27 equals 3

a to the power of negative 1 over n end exponent equals open parentheses a to the power of 1 over n end exponent close parentheses to the power of negative 1 end exponent equals open parentheses n-th root of a close parentheses to the power of negative 1 end exponent equals fraction numerator 1 over denominator n-th root of a end fraction

A negative, fractional power is one over a root

64 to the power of negative 1 half end exponent equals fraction numerator 1 over denominator square root of 64 end fraction equals 1 over 8

125 to the power of negative 1 third end exponent equals fraction numerator 1 over denominator cube root of 125 end fraction equals 1 fifth

a to the power of m over n end exponent equals a to the power of 1 over n cross times m end exponent equals open parentheses a to the power of 1 over n end exponent close parentheses to the power of m equals open parentheses a to the power of m close parentheses to the power of 1 over n end exponent

The fractional power m over n is the nth root all to the power m, open parentheses n-th root of blank close parentheses to the power of m, or the nth root of the power m, n-th root of open parentheses blank close parentheses to the power of m end root (both are the same)

8 to the power of 2 over 3 end exponent equals open parentheses 8 to the power of 1 third end exponent close parentheses squared equals open parentheses cube root of 8 close parentheses squared equals 2 squared equals 4

8 to the power of 2 over 3 end exponent equals open parentheses 8 squared close parentheses to the power of 1 third end exponent equals cube root of 64 equals 4

Examiner Tips and Tricks

The index laws are not in the formula booklet so you must remember them!

How do I change the base?

  • Index laws only work with terms that have the same base

    • 2 cubed cross times 5 squared cannot be simplified using index laws

  • You can sometimes rewrite a base as a power of another base

    • 2 to the power of 5 cross times bold 4 cubed equals 2 to the power of 5 cross times open parentheses bold 2 to the power of bold 2 close parentheses cubed

      • The 4 changes to open parentheses 2 squared close parentheses

      • This is called changing the base

    • It can now be simplified using index laws

    • 2 to the power of 5 cross times open parentheses 2 squared close parentheses cubed equals 2 to the power of 5 cross times 2 to the power of 6 equals 2 to the power of 11

Worked Example

Simplify the following expressions:

i) fraction numerator left parenthesis 3 x squared right parenthesis left parenthesis 2 x cubed y squared right parenthesis over denominator left parenthesis 6 x squared y right parenthesis end fraction

 

ai-sl-1-1-2-laws-of-indices-we-i

ii) left parenthesis 4 x squared y to the power of negative 4 end exponent right parenthesis cubed left parenthesis 2 x cubed y to the power of negative 1 end exponent right parenthesis to the power of negative 2 end exponent

 

ai-sl-1-1-2-laws-of-indices-we-ii


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