Rules of Indices

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Indices Rules

Indices Rules builds on the 7 rules from Powers and Roots. We will cover 3 more complicated rules here. Make sure you are confident with the following topics before moving onto laws and indices.

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Indices Rule 8: Fractional Powers

The fractional indices laws apply when the power is a fraction.

abc=abctextcolor{red}{a}^{large{frac{textcolor{blue}{b}}{textcolor{limegreen}{c}}}} = sqrt[textcolor{limegreen}{c}]{textcolor{red}{a}^textcolor{blue}{b}}

This is commonly use to show square and cube roots.

x12=x12=x2textcolor{red}{x}^{large{frac{textcolor{limegreen}{1}}{textcolor{blue}{2}}}}= sqrt[textcolor{blue}{2}]{textcolor{red}{x}^textcolor{limegreen}{1}} =sqrt[textcolor{blue}{2}]{textcolor{red}{x}}

x13=x13=x3textcolor{red}{x}^{large{frac{textcolor{limegreen}{1}}{textcolor{blue}{3}}}}= sqrt[textcolor{blue}{3}]{textcolor{red}{x}^textcolor{limegreen}{1}} =sqrt[textcolor{blue}{3}]{textcolor{red}{x}}

Note: it doesn’t matter which order you carry out the square root and multiplication operations. In other words, the rule can also be written as

abc=(ac)btextcolor{red}{a}^{large{frac{textcolor{blue}{b}}{textcolor{limegreen}{c}}}} = (sqrt[textcolor{limegreen}{c}]{textcolor{red}{a}})^textcolor{blue}{b}

You should try to carry out the operations in the order that makes the calculation as simple as possible.

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Indices Rule 9: Multi-step Fractional Powers

You may also be asked to simplify expressions where the numerator is not 1bf{1}.

6423=6423textcolor{red}{64}^{large{frac{textcolor{limegreen}{2}}{textcolor{blue}{3}}}}= sqrt[textcolor{blue}{3}]{textcolor{red}{64}^textcolor{limegreen}{2}}

643=4sqrt[textcolor{blue}{3}]{textcolor{red}{64}} = textcolor{red}{4}

42=16textcolor{red}{4}^textcolor{limegreen}{2} = textcolor{red}{16}

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Indices Rule 10: Negative Powers

Negative powers flip the fraction and put 11 over the number

In general, the result of a negative power is “1bf{1} over that number to the positive power”, i.e.

a−b=1abtextcolor{red}{a}^{-textcolor{limegreen}{b}} = dfrac{1}{textcolor{red}{a}^textcolor{limegreen}{b}}

for any value of aa or bb. When the power is −1textcolor{blue}{-1}, this takes the form,

a−1=1atextcolor{red}{a}^{textcolor{blue}{-1}}=dfrac{1}{textcolor{red}{a}} or 10−1=110textcolor{red}{10}^{textcolor{blue}{-1}} = dfrac{1}{textcolor{red}{10}}

When the number is a fraction, the negative power flips the fraction.

(ab)−x=(ba)xbigg(dfrac{textcolor{blue}{a}}{textcolor{limegreen}{b}}bigg)^{-textcolor{red}{x}} = bigg(dfrac{textcolor{limegreen}{b}}{textcolor{blue}{a}}bigg)^textcolor{red}{x} 

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Example 1: Negative Powers

Simplify the following,  4−34^{-3}.

[2 marks]

We now know that 4−34^{-3} is equal to 143dfrac{1}{4^3}. We also know that

43=4×4×4=16×4=644^3=4times 4times 4=16times 4=64.

So, we get that

4−3=1644^{-3}=frac{1}{64}.

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Example 2: Fractional Powers and Roots

Simplify the following,  9329^{frac{3}{2}}.

[2 marks]

So, we know that 9329^{frac{3}{2}} is equal to 932sqrt[2]{9^3} or (92)3(sqrt[2]{9})^3.

So, to work out (92)3(sqrt[2]{9})^3, we first have to square root 99, which is easy enough – the square root of 99 is 33. So, (92)3(sqrt[2]{9})^3 becomes 333^3, which is

33=3×3×3=273^3=3times 3times 3 = 27

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Example 3: Multiplication and Powers

Write 215×8−42^{15}times 8^{-4} as a power of 22, and hence evaluate the expression. (Non calculator)

[3 marks]

The first part of the expression is a power of 22, whilst the second part is a power of 88.

we know that

8=238 = 2^3

This means we can rewrite the following,

8−4=(23)−48^{-4}=left(2^3right)^{-4}

Next, using Rule 3, we can simplify,

(23)−4=23×(−4)=2−12left(2^3right)^{-4}=2^{3times(-4)}=2^{-12}

So the whole expression can be written as

215×2−12,2^{15}times2^{-12},

Finally using Rule 1 we simplify the expression further.

215×2−12=215+(−12)=232^{15}times2^{-12}=2^{15+(-12)}=2^3

Thus, we have written the expression as a power of 22. Evaluating this final answer gives

23=82^3 = 8

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Rules of Indices Example Questions

Question 1: Write 95×3−59^5times3^{-5} as a power of 33

[3 marks]

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So, we can’t use any laws straight away since the terms don’t have the same base. However, if we recognise that 9=329=3^2, then we can write the first term as

 

(32)5left(3^2right)^5

 

Using the power law, we get

 

(32)5=32×5=310left(3^2right)^5=3^{2times5}=3^{10}

 

Therefore, the whole expression becomes

 

310×3−53^{10}times3^{-5}

 

Applying the multiplication law, this simplifies to

 

310+(−5)=353^{10+(-5)}=3^5

 

Thus, we have written the expression as a power of 33.

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Question 2: Work out 9×6−2sqrt{9}times 6^{-2}

Write your answer in its simplest form (Non-calculator)

[3 marks]

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Firstly, as 32=93^2=9, the inverse operation gives, 9=3sqrt{9}=3

 

So, that leaves 6−26^{-2}, this becomes the following fraction,

 

 6−2=1626^{-2}=dfrac{1}{6^2}

 

We know that 62=6×6=366^2=6times 6=36, so

 

6−2=1366^{-2}=dfrac{1}{36}

 

Multiplying our two answers together, we get

 

9×6−2=3×136=336=112sqrt{9}times 6^{-2}=3timesdfrac{1}{36}=dfrac{3}{36}=dfrac{1}{12}

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Question 3: Work out 412×4324^{frac{1}{2}}times4^{frac{3}{2}}

[3 marks]

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This expression can be rewritten as,

 

4×(4)3sqrt4 times (sqrt4)^3

 

Given we know that 4=2sqrt4=2 , this becomes,

2×232times2^3

 

Hence,

 

2×23=2×8=162times2^3=2times8=16

 

Notice that in this example we chose to perform the 4sqrt{4} operation before cubing the answer. We could alternatively write the expression as 43sqrt{4^3}, but in this case the first option is easier.

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Question 4: Work out 8−538^{-frac{5}{3}} (HIGHER ONLY)

(Non-calculator)

[3 marks]

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As it is a negative power we can rewrite this as,

 

8−53=18538^{-frac{5}{3}}=frac{1}{8^{frac{5}{3}}}

 

Now, we can work out the denominator, which we will write as,

 

853=853=(83)58^{frac{5}{3}}=sqrt[3]{8^5}=(sqrt[3]{8})^5

 

We know that 83=2sqrt[3]{8}=2. So this simplifies to,

 

(83)5=25(sqrt[3]{8})^5=2^5

 

Counting up in powers of 22: 44, 88, 1616, 3232 – we see that 3232 is the 55th power of 22, so

 

835=32sqrt[3]{8}^5=32

 

Therefore, the answer is,

 

8−53=1328^{-frac{5}{3}}=dfrac{1}{32}

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Specification Points Covered

Algebra – 4. simplify and manipulate algebraic expressions (including those involving surds and algebraic fractions) by:

  • collecting like terms
  • multiplying a single term over a bracket
  • taking out common factors
  • expanding products of two or more binomials
  • factorising quadratic expressions of the form x2+bx+cx^2 + bx + c, including the
  • difference of two squares; factorising quadratic expressions of the form ax2+bx+cax^2 + bx + c
  • simplifying expressions involving sums, products and powers, including the laws of indices

Rules of Indices Worksheet and Example Questions

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(NEW) Rules of Indices Exam Style Questions - MME

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Rules of Indices Drill Questions

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Indices Rules - Drill Questions

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Fractional And Negative Indices - Drill Questions

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Rules of Indices - Drill Questions

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Related Topics

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Collecting Like Terms

Level 1-3GCSEKS3
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Powers and Roots

Level 4-5GCSEKS3
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