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Laws of Indices (Cambridge (CIE) IGCSE International Maths)

Revision Note

Jamie Wood

Written by: Jamie Wood

Reviewed by: Dan Finlay

Cambridge (CIE) International Maths: ExtendedEdexcel Maths A (Modular): Higher Unit 1Edexcel Maths A: HigherCambridge (CIE) Maths: ExtendedCambridge (CIE) International Maths: CoreCambridge (CIE) Maths: Core

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Laws of Indices

What are the laws of indices?

  • Index laws are rules you can use when doing operations with powers

    • They work with 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

How do I deal with different bases?

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

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

  • Sometimes expressions involve different base values, but one is related to the other by a power

    • e.g. 2 to the power of 5 cross times 4 cubed

  • You can use powers to rewrite one of the bases

    • 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

    • This can then be simplified more easily, as the two bases are now the same

    • 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

(a) Find the value of x when 6 to the power of 10 space cross times space 6 to the power of x space equals space 6 squared

Using the law of indices 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 we can rewrite the left hand side

 6 to the power of 10 cross times 6 to the power of x equals 6 to the power of 10 plus x end exponent

So the equation is now

6 to the power of 10 plus x end exponent equals 6 squared

Comparing both sides, the bases are the same, so we can say that

10 plus x equals 2

Subtract 10 from both sides

bold italic x bold equals bold minus bold 8

(b) Find the value of n when 5 to the power of n divided by 5 to the power of 4 equals 5 to the power of 6

Using the law of indices a to the power of m divided by a to the power of n equals a to the power of m minus n end exponent we can rewrite the left hand side

5 to the power of n divided by 5 to the power of 4 equals 5 to the power of n minus 4 end exponent

So the equation is now 

5 to the power of n minus 4 end exponent equals 5 to the power of 6

Comparing both sides, the bases are the same, so we can say that

n minus 4 equals 6

Add 4 to both sides

bold italic n bold equals bold 10

 

(c) Without using a calculator, find the value of 2 to the power of negative 4 end exponent

Using the law of indices a to the power of negative n end exponent equals 1 over a to the power of n we can rewrite the expression

2 to the power of negative 4 end exponent equals 1 over 2 to the power of 4

2 to the power of 4 equals 2 cross times 2 cross times 2 cross times 2 equals 16 so we can rewrite the expression

1 over 2 to the power of 4 equals 1 over 16

bold 1 over bold 16

(d) Without using a calculator, find the value of 8 to the power of negative 1 third end exponent

Using the law of indices a to the power of negative n end exponent equals 1 over a to the power of n we can rewrite the expression

8 to the power of negative 1 third end exponent equals 1 over 8 to the power of 1 third end exponent

Using the law of indices a to the power of 1 over n end exponent equals n-th root of a we can rewrite the expression

1 over 8 to the power of 1 third end exponent equals fraction numerator 1 over denominator cube root of 8 end fraction

The cube root of 8 is 2

1 half

(e) Without using a calculator, find the value of 81 to the power of 3 over 4 end exponent.

Use the law of indices 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 we can rewrite the expression in two ways

81 to the power of 3 over 4 end exponent equals open parentheses 81 cubed close parentheses to the power of 1 fourth end exponent or open parentheses 81 to the power of 1 fourth end exponent close parentheses cubed

Both forms are equivalent, but open parentheses 81 cubed close parentheses to the power of 1 fourth end exponent would require calculating 81 cubed, so use the second form instead

Using the law of indices a to the power of 1 over n end exponent equals n-th root of a we can rewrite the expression

open parentheses 81 to the power of 1 fourth end exponent close parentheses cubed equals open parentheses fourth root of 81 close parentheses cubed

The 4th root of 81 is 3 as 3×3×3×3=34=81

open parentheses 3 close parentheses cubed

Lastly, calculate or recall 3 cubed

27

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    Author
    Reviewer
    Jamie Wood

    Author: Jamie Wood

    Expertise: Maths

    Jamie graduated in 2014 from the University of Bristol with a degree in Electronic and Communications Engineering. He has worked as a teacher for 8 years, in secondary schools and in further education; teaching GCSE and A Level. He is passionate about helping students fulfil their potential through easy-to-use resources and high-quality questions and solutions.

    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.