1. Indices
Indicial notation is frequently used in mathematics and you must understand and be able to use it. This chapter discusses:
- Index notation
- Whole number indices
- Fractional indices
Do you need this chapter?
Here are some questions on the topics in this chapter to consider. If you can do these you may skip this chapter.
Quiz: Indices
1. Simplify
[latex]\qquad\qquad x^4×x^2[/latex]
2. Simplify
[latex]\qquad\qquad\dfrac{y^7}{y^3}[/latex]
3. Simplify
[latex]\qquad\qquad\left(z^7\right)^{3}[/latex]
4. Simplify
[latex]\qquad\qquad\dfrac{15a^4b}{6a^2b^2}[/latex]
5. Simplify and express with positive indices
[latex]\qquad\qquad\dfrac{12m^{-4}n}{3m^{-5}n^{-2}}[/latex]
6. Simplify and give answers with positive indices
[latex]\qquad\qquad\dfrac{\left( 25p{^4}q^2 \right)^{1/2}}{\left( pq \right)^{-1/2}}[/latex]
Answers
[latex]\qquad \qquad \mathbf{1.}\ x^6[/latex]
[latex]\qquad \qquad \mathbf{2.}\ y^4[/latex]
[latex]\qquad \qquad \mathbf{3.}\ z^{21}[/latex]
[latex]\qquad \qquad \mathbf{4.}\ \dfrac{5a^2}{2b}[/latex]
[latex]\qquad \qquad \mathbf{5.} \ 4mn^3[/latex]
[latex]\qquad \qquad \mathbf{6.} \ 5p^{\frac{5}{2}} q^\frac{3}{2}[/latex]
If you need to review this topic, continue reading.
For a quick introduction, please watch the following video. Further details are given in the following section
Indices (9:09 min)
Indices (9:09 min) by RMIT University Library Videos (YouTube)
Index notation
Indicial notation is a shorthand way of writing multiple products.
Consider the following examples:
Examples: Index notation
1. [latex]\ 3^{4} =3\times3\times3\times3=81[/latex].
2. [latex]\ 5^{3} =5\times5\times5=125[/latex].
3. [latex]\ 2^{7} =2\times2\times2\times2\times2\times2\times2=128[/latex].
In general:
[latex]a^{n}=\underbrace{a\times a\times a\times\ldots\times a}_{n\textrm{ factors}}[/latex]
The letter [latex]n[/latex] in [latex]a^{n}[/latex] is referred to in one of three ways:
- [latex]n[/latex] is the index in [latex]a^{n}[/latex] with [latex]a[/latex] known as the base.
- [latex]n[/latex] is the exponent or power to which the base [latex]a[/latex] is raised.
- [latex]n[/latex] is the logarithm, with [latex]a[/latex] as the base.
When a number such as [latex]125[/latex] is written in the form [latex]5^{3}[/latex] we say
it is written as an exponential or in index notation. Multiplication
and division of numbers or expressions written in index notation is
achieved using index laws.
Index laws
This section states and gives examples of universal index laws.
1. Multiplication
To multiply index expressions with the same base you add the indices:
[latex]a^{m}\times a^{n}=a^{m+n}[/latex]
Example: Index law 1
[latex]\begin{align*} 2^{3}\times2^{2} & =\left(2\times2\times2\right)\times\left(2\times2\right)\\ & =2\times2\times2\times2\times2\\ & =2^{5} \end{align*}[/latex]
Therefore [latex]2^{3}\times2^{2}=2^{3+2}=2^{5}[/latex].
2. Division
To divide expressions with the same base, subtract the indices:
[latex]a^{m}\div a^{n}=\dfrac{a^m}{a^n}=a^{m-n}.[/latex]
Example: Index law 2
[latex]\begin{align*} 3^{5}\div3^{3} &=\frac{3^{5}}{3^{3}}\\ &=\frac{3\times3\times3\times3\times3}{3\times3\times3}\\ &=\frac{3\times3}{1}\quad\textrm{cancelling three lots of 3}\\ &=3^{2}. \end{align*}[/latex]
Therefore [latex]3^{5}\div3^{3}=\dfrac{3^{5}}{3^{3}}=3^{5-3}=3^{2}[/latex].
Note that expressions in index form can only be multiplied or divided if they have the same base.
3. Index form raised to a further power
To raise an expression in index form to a power, multiply the indices:
In general, we have the third index law:
[latex]\left(a^{m}\right)^{n}=a^{m\times n}.[/latex]
This also leads to the expression:
[latex]\begin{align*} (a^{m}b^{p})^{n} & =a^{mn}b^{pn}. \end{align*}[/latex]
Be careful as this is true for multiplication and division only,
not addition or subtraction , so that [latex](a+b)^{n}\neq a^{n}+b^{n}[/latex].
Example: Index law 3
[latex]\begin{align*} \left(5^{2}\right)^{3} & =5^{2}\times5^{2}\times5^{2}\\ & =5^{2+2+2}\quad\textrm{using the first index law}\\ & =5^{6}. \end{align*}[/latex]
Therefore [latex]\left(5^{2}\right)^{3}=5^{2\times3}=5^{6}[/latex].
Examples: Index laws 1-3
1. Simplify [latex]x^{5}\times x^{6}[/latex].
Solution:
[latex]\begin{align*} x^{5}\times x^{6} & =x^{5+6}\quad\textrm{using the first index law}\\ & =x^{11}. \end{align*}[/latex]
2. Simplify [latex]a^{5}\div a^{3}[/latex].
Solution:
[latex]\begin{align*} a^{5}\div a^{3} & =\frac{a^{5}}{a^{3}}=a^{5-3}\quad\textrm{by the second law}\\ & =a^{2}. \end{align*}[/latex]
3. Simplify [latex]\left(c^{3}\right)^{4}[/latex].
Solution:
[latex]\begin{align*} \left(c^{3}\right)^{4} & =c^{3\times4}\quad\textrm{by the third law}\\ & =c^{12}. \end{align*}[/latex]
4. Simplify [latex]\left(2x^{2}\right)^{3}[/latex].
Solution:
[latex]\begin{align*} \left(2x^{2}\right)^{3} & =2^{3}\left(x^{2}\right)^{3}\\ & =8x^{2\times3}\quad\textrm{by the third law}\\ & =8x^{6}. \end{align*}[/latex]
Try these exercises.
Exercises: Index laws 1-3
Simplify the following:
[latex]\begin{array}{lllll} \qquad \mathbf{1.} \ c^{5}\times c^{3}\times c^{7} & & \mathbf{2.} \ 3\times2^{2}\times2^{3} & & \mathbf{3.} \ a^{3}\times a^{2}b^{3}\times ab^{4}\\ & & & & \\ \qquad \mathbf{4.} \ 3^{6}\div3^{4} & & \mathbf{5.} \ a^{8}\div a^{3} & & \mathbf{6.}\ x^{4}y^{6}\div x^{2}y^{3}\\ \qquad \mathbf{7.}\ \left(x^{3}\right)^{4} & & \mathbf{8.} \ \left(x^{m}y^{n}\right)^{5} & & \\ \end{array}[/latex]
Answers
[latex]\begin{array}{llll} \qquad \mathbf{1.}\ c^{15} & \mathbf{2.}\ 3\times2^{5}=96 & \mathbf{3.}\ a^{6}b^{7} & \mathbf{4.}\ 3^{2}=9\\ \qquad \mathbf{5.}\ a^{5} & \mathbf{6.}\ x^{2}y^{3} &\mathbf{7.}\ x^{12} & \mathbf{8.}\ x^{5m}y^{5n} \end{array}[/latex]
The index laws also apply if the index is zero, negative or a fraction.
4. Zero index
Consider [latex]2^{3}\div2^{3}=\dfrac{2^{3}}{2^{3}}=\dfrac{2\times2\times2}{2\times2\times2}=\dfrac{8}{8}=8\div8=1[/latex].
Using the second law, [latex]2^{3}\div2^{3}=2^{3-3}=2^{0}[/latex] therefore [latex]1=2^{0}[/latex].
In general, any expression with a zero index is equal to 1. That is
[latex]a^0=1,\quad a\neq0.[/latex]
Note that [latex]0^{0}[/latex] is ambiguous so we don't allow [latex]a=0[/latex]. We will call this the zero index law.
Examples: Index law 4
1. [latex]7^{0}=1[/latex]
2. [latex]\left(xy\right)^{0}=1[/latex]
3. [latex]\left(\dfrac{1}{2}\right)^{0}=1[/latex]
4. [latex]\left(28x^{2}\right)^{0}=1[/latex]
5. Negative indices
Consider [latex]2^{0}\div2^{4}[/latex].
[latex]\begin{align*} 2^{0}\div2^{4} & =\frac{2^{0}}{2^{4}}\quad\textrm{remember that } 2^0=1\\ & =\frac{1}{2^{4}}. \end{align*}[/latex]
But
[latex]\begin{align*} 2^{0}\div2^{4} & =2^{0-4}\quad\textrm{using the second law}\\ & =2^{-4}. \end{align*}[/latex]
So [latex]2^{0}\div2^{4}=2^{-4}[/latex] and [latex]2^{0}\div2^{4}=\dfrac{1}{2^{4}}[/latex]. Therefore [latex]2^{-4}=\dfrac{1}{2^{4}}[/latex].
In general,
[latex]a^{-n}=\dfrac{1}{a^n},\qquad a\neq 0[/latex]
which also leads to
[latex]\dfrac{1}{a^{-n}}=a^{n},\qquad a\neq 0.[/latex]
We will call this the negative index law.
Examples: Index law 5
1. [latex]2^{-3} =\dfrac{1}{2^3} =\dfrac{1}{8}.[/latex]
2. [latex]\dfrac{1}{x}=x^{-1}.[/latex]
3. [latex]2y^{-1}=\dfrac{2}{y}.[/latex]
4. [latex]\dfrac{1}{3x^{-2}}=\dfrac{x^{2}}{3}.[/latex]
5. [latex]\dfrac{1}{\left(-2a\right)^{-3}}=\left(-2a\right)^{3} = -8a^3.[/latex]
6. [latex]5ab^{-4}=\dfrac{5a}{b^{4}}.[/latex]
7. [latex]5 \left(ab \right)^{-4}=\dfrac{5}{ \left( ab \right) ^{4}}.[/latex]
8. [latex]5a^{-4}b=\dfrac{5b}{a^{4}}.[/latex]
6. Fractional indices
What meaning can we give to [latex]3^{1/2}[/latex] ? Using the first index law, we find:
[latex]\begin{align*} 3^{1/2}\times3^{1/2} & =3^{1/2+1/2}\\ & =3^{1}\\ & =3. \end{align*}[/latex]
So [latex]3^{1/2}[/latex] is the number that when multiplied by itself, gives three. But the square root of [latex]3[/latex] is the number that when multiplied by itself gives [latex]3[/latex]. That is:
[latex]\begin{align*} \sqrt{3}\times\sqrt{3} & =3. \end{align*}[/latex]
Since [latex]3^{1/2}[/latex] behaves like [latex]\sqrt{3}[/latex] we say
[latex]3^{1/2}=\sqrt{3}.[/latex]
Now consider [latex]2^{1/3}[/latex]. Using the first index law, we can write:
[latex]\begin{align*} 2^{1/3}\times2^{1/3}\times2^{1/3} & =2^{1/3+1/3+1/3}\\ & =2^{1}\\ & =2. \end{align*}[/latex]
So [latex]2^{1/3}[/latex] behaves like the cube root of 2. That is,
[latex]\begin{align*} 2^{1/3} & =\sqrt[3]{2}. \end{align*}[/latex]
In general, we say that [latex]a^{1/n}[/latex] is the [latex]n^\mathrm{th}[/latex] root of [latex]a[/latex]. That is
[latex]a^{1/n}=\sqrt[n]{a}[/latex]
for any positive integer [latex]n[/latex] (whole numbers like [latex]1,2,\ldots[/latex] ).
Examples: Index law 6
1. [latex]4^{1/2}=\sqrt{4}=2[/latex].
2. [latex]27^{1/3}=\sqrt[3]{27}=3[/latex].
3. [latex]3^{1/4}=\sqrt[4]{3}[/latex].
4. [latex]b^{1/5}=\sqrt[5]{b}[/latex].
5. [latex]x^{1/2}=\sqrt{x}[/latex].
6. [latex]32^{-1/5}=\dfrac{1}{32^{1/5}}=\dfrac{1}{\sqrt[5]{32}}=\dfrac{1}{2}[/latex].
In most cases, the root of a number will not be able to be written as a whole number or fraction and will be an irrational number. For example, [latex]\sqrt{2}=1.41421356\ldots[/latex].
Here are some exercises for you to try.
Exercises: Index laws 4 - 6
Write with positive indices and evaluate if possible:
[latex]\begin{array}{lllllll} \qquad \mathbf{1.}\ x^{-6} & & \mathbf{2.} \ 250^{0} & & \mathbf{3.}\ 3ab^{-5} & & \mathbf{4.} \ \left(pq\right)^{-2}\\ & & & & & & \\ \qquad \mathbf{5.}\ \left(5xy\right)^{-3} & & \mathbf{6.} \ \dfrac{2y}{z^{-5}} & & \mathbf{7.} \ 2^{-5} & & \mathbf{8.}\ (-2)^{-3}\\ & & & & & & \\ \qquad \mathbf{9.}\ -(3^{-2}) & &\mathbf{10.} \ 2\times(-5)^{-2} &&\mathbf{11.}\ 64^{1/2}&&\mathbf{12.}\ 125^{1/3}\\ &&& & & & \\ \qquad13.\ 36^{1/4}&&14.\ 81^{-1/2}&&15.\ 128^{-1/7}\quad\quad\\ \end{array}[/latex]
[latex]\qquad \qquad[/latex]
Answers
[latex]\begin{array}{lllll} \qquad \mathbf{1.}\ \dfrac{1}{x^{6}} & 2.\mathbf{2.}\ 1 & \mathbf{3.}\ \dfrac{3a}{b^{5}}. & \mathbf{4.}\ \dfrac{1}{(pq)^{2}} & \mathbf{5.}\ \dfrac{1}{(5xy)^{3}}=\dfrac{1}{125x^{3}y^{3}}\\ \\ \qquad \mathbf{6.}\ 2yz^{5} & \mathbf{7.}\ \dfrac{1}{32} & \mathbf{8.}\ -\dfrac{1}{8} & \mathbf{9.}\ -\dfrac{1}{9} & \mathbf{10.}\ \dfrac{2}{25}\\ \\ \qquad \mathbf{11.}\ 8 & \mathbf{12.}\ 5 & \mathbf{13.}\ ≈ 2.45 & \mathbf{14.}\ \dfrac{1}{9} & 15.\ \dfrac{1}{2} \end{array}[/latex]
Combining index laws
The index laws are very useful for simplifying complex expressions. Frequently you will apply combinations of the index laws to achieve a desired result. Some examples follow.
Examples: Combining index laws
1. Simplify [latex]\left(4a^{2}b\right)^{3}\div b^{2}[/latex].
Solution:
[latex]\begin{align*} \left(4a^{2}b\right)^{3}\div b^{2} & =\left(4^{3}a^{6}b^{3}\right)\div b^{2}\quad\textrm{using law 3}\\ & =4^{3}a^{6}b^{1}\quad\textrm{using law 2}\\ & =4^{3}a^{6}b\\ & =64a^{6}b. \end{align*}[/latex]
2. Simplify[latex]\left(\dfrac{3a^{3}b}{c^{2}}\right)^{2}\div\left(\dfrac{ab}{3c^{-2}}\right)^{-3}.[/latex]
Solution:
[latex]\begin{align*} \left(\frac{3a^{3}b}{c^{2}}\right)^{2}\div\left(\frac{ab}{3c^{-2}}\right)^{-3} & =\frac{3^{2}a^{6}b^{2}}{c^{4}}\div\frac{a^{-3}b^{-3}}{3^{-3}c^{6}}\quad\textrm{by law 3}\\ & =\frac{3^{2}a^{6}b^{2}}{c^{4}}\times\frac{3^{-3}c^{6}}{a^{-3}b^{-3}}\\ & =\frac{3^{2-3}a^{6}b^{2}c^{6}}{c^{4}a^{-3}b^{-3}}\quad\textrm{by law 1}\\ & =3^{-1}a^{6-\left(-3\right)}b^{2-\left(-3\right)}c^{6-4}\quad\textrm{by law 2}\\ & =3^{-1}a^{9}b^{5}c^{2}\quad\textrm{simplifying}\\ & =\frac{a^{9}b^{5}c^{2}}{3}\quad\textrm{by negative index law.} \end{align*}[/latex]
3. Write [latex]x^{-1}+x^{2}[/latex] as a single fraction.
Solution:
[latex]\begin{align*} x^{-1}+x^{2} & =\frac{1}{x}+x^{2}\quad\textrm{by negative index law}\\ & =\frac{1}{x}+\frac{xx^{2}}{x}\\ & =\frac{1+xx^{2}}{x}\quad\textrm{using a common denominator}\\ & =\frac{1+x^{3}}{x}\quad\textrm{using law 1.} \end{align*}[/latex]
Try the following exercises on your own.
Exercises: Combining index laws
Simplify the following:
[latex]\begin{array}{lllll} \qquad \mathbf{1.}\ 2a^{3}b^{2}\times a^{-1}\times b^{3} & & \mathbf{2.}\ (5x^{-2}y)^{-3} & & \mathbf{3.} \ (3x^{3}y^{-1})^{5}\\ & & & & \\ \qquad \mathbf{4.}\ (a^{-4}b^{-5})^{-2} & & \mathbf{5.}\ \dfrac{a^{2}b^{3}c^{-4}}{a^{4}bc^{5}} & & \mathbf{6.}\ \dfrac{a^{7}\times a^{8}\times a^{3}}{a^{2}\times a^{5}}\\ & & & & \\ \qquad\mathbf{7.}\ x\left(x-x^{-1}\right) & & \mathbf{8.}\ \dfrac{(2^{4})^{n}}{2^{3}} & &\mathbf{9.}\ \dfrac{15a^{2}b}{3a^{4}b}\times\dfrac{4a^{5}b^{2}}{5a^{3}b^{4}}\\ & & & & \\ \qquad\mathbf{10.}\ 2^{4}-2^{3} & & & & \\ \end{array}[/latex]
Answers
[latex]\begin{array}{lllllc} \mathbf{1.}\;2a^{2}b^{5} & \mathbf{2.}\;\dfrac{x^{6}}{5^{3}y^{3}} &\mathbf{3.}\;\dfrac{3^{5}x^{15}}{y^{5}} & \mathbf{4.}\;a^{8}b^{10} \\ \\ \mathbf{5.}\;\dfrac{b^{2}}{a^{2}c^{9}}&\mathbf{6.}\;a^{11} & \mathbf{7.}\;x^{2}-1 & \mathbf{8.}\;2^{4n-3} \\ \\ \mathbf{9.}\;\dfrac{4a^{7}b^{3}}{a^{7}b^{5}}=4b^{-2}=\dfrac{4}{b^{2}} & \mathbf{10.}\; 2^{3}(2^{1}-2^{0})=2^{3}=8 \end{array}[/latex]
More complex fractional indices
We now consider terms like [latex]a^{2/3}[/latex].
Recall the third index law: [latex]\left(a^{m}\right)^{n}=a^{mn}[/latex]. Hence
[latex]\begin{align*} a^{2/3} & =\left(a^{2}\right)^{1/3}\\ & =\sqrt[3]{a^{2}}\quad\textrm{using the definition of indices as roots.} \end{align*}[/latex]
Again using the third index law we can write:
[latex]\begin{align*} a^{2/3} & =a^{2\times\left(1/3\right)}\\ & =a^{\left(1/3\right)\times2}\\ & =\left(\sqrt[3]{a}\right)^{2}. \end{align*}[/latex]
In general,
[latex]\begin{align*} a^{p/q} & =\sqrt[q]{a^{p}} \end{align*}[/latex]
or
[latex]\begin{align*} a^{p/q} & =\left(\sqrt[q]{a}\right)^{p} \end{align*}[/latex]
where [latex]p[/latex] and [latex]q[/latex] are integers with [latex]q\neq0[/latex].
Examples: More complex fractional indices
1. [latex]5^{3/4}=\sqrt[4]{5^{3}}[/latex] or [latex]\left(\sqrt[4]{5}\right)^{3}[/latex]
2. [latex]7^{5/2} = \sqrt{7^{5}} = \left(\sqrt{7}\right)^{5}[/latex]
3. [latex]a^{7/5} = \sqrt[5]{a^{7}}=\left(\sqrt[5]{a}\right)^{7}[/latex]
4. [latex]y^{-\frac{3}{4}} = \dfrac{1}{y^{3/4}}=\dfrac{1}{\sqrt[4]{y^{3}}}[/latex] or [latex]\dfrac{1}{\left(\sqrt[4]{y}\right)^{3}}[/latex]
5. [latex]\sqrt[4]{x^{3}}=x^{3/4}[/latex] or [latex]\left( \sqrt[4]{x} \right)^3[/latex]
Simplification of expressions
As shown in the following examples, the index laws may be used to simplify algebraic and numerical expressions. Note that, in practice, you don't have to put in the lines of text explaining each step. They are only shown for explanation.
Examples: Simplification of expressions
1. Simplify [latex]\left(x^{3}y^{9}\right)^{2/3}[/latex].
Solution:
[latex]\begin{align*} \left(x^{3}y^{9}\right)^{2/3} & =x^{3\times\left(2/3\right)}y^{9\times\left(2/3\right)}\quad\textrm{using the third law,}\\ & =x^{2}y^{6}. \end{align*}[/latex]
2. Simplify [latex]3^{1/3}\div3^{4/3}[/latex].
Solution:
[latex]\begin{align*} 3^{1/3}\div3^{4/3} & =3^{\left(1/3\right)-\left(4/3\right)}\quad\textrm{using the second index law,}\\ & =3^{-\left(3/3\right)}\\ & =3^{-1}\\ & =\frac{1}{3}. \end{align*}[/latex]
3. Simplify [latex]32^{3/5}[/latex].
Solution:
For this example, you need to know the fifth root of [latex]32[/latex] is [latex]2[/latex].
[latex]\begin{align*} 32^{3/5} & =\left(\sqrt[5]{32}\right)^{3}\\ & =2^{3}\\ & =8. \end{align*}[/latex]
4. Simplify [latex]25^{-1/2}[/latex].
Solution:
For this example, you need to know the square root of [latex]25[/latex] is [latex]5[/latex]. You also need to use index law [latex]5[/latex]: [latex]a^{-n}=\frac{1}{a^{n}}.[/latex]
Then
[latex]\begin{align*}25^{-1/2} & =\frac{1}{\sqrt{25}}\quad\textrm{using index law 5,}\\& =\frac{1}{5}.\end{align*}[/latex]
5. Simplify [latex]\left(a^{2}b^{5}\right)^{1/3}\times a^{1/3}b^{-2/3}[/latex].
Solution:
[latex]\begin{align*} \left(a^{2}b^{5}\right)^{1/3}\times a^{1/3}b^{-2/3} & =a^{2/3}b^{5/3}\times a^{1/3}b^{-2/3}\quad\textrm{using the third index law,}\\ & =a^{\left(2/3\right)+\left(1/3\right)}b^{\left(5/3\right)-\left(2/3\right)}\\ & =ab. \end{align*}[/latex]
Please do the following exercises for some practice.
Exercises: Simplification of expressions
Simplify the following expressions. Give your answers in index notation with positive indices.
1. [latex]\left(\dfrac{8}{27}\right)^{2/3}[/latex]
2. [latex]\sqrt{5}\times\sqrt[3]{5}\times\sqrt[6]{5}[/latex]
3. [latex]\sqrt{a}\times\sqrt[4]{a}\times\sqrt[3]{a^{2}}[/latex]
4. [latex]\left(125a^{6}b\right)^{1/3}\times b^{2/3}[/latex]
5. [latex]\dfrac{\left(2xy^{3}\right)^{1/2}}{2}\times\left(\dfrac{x^{3/2}}{y^{2}}\right)^{4}[/latex]
6. [latex]2^{5/2}-2^{3/2}[/latex]
Answers
[latex]\begin{array}{lllllllllll} \qquad \mathbf{1.}\ \dfrac{2^{2}}{3^{2}} & & \mathbf{2.}\ 5 & & \mathbf{3.}\ \;a^{17/12} & & \mathbf{4.}\ \;5a^{2}b & & \mathbf{5.}\ \dfrac{x^{13/2}}{2^{1/2}y^{13/2}} & & \mathbf{6.}\ 2^{3/2}\end{array}[/latex]
Summary of index laws
Key takeaways: Indices and index laws
1. First Law
[latex]\ a^{m}\times a^{n}=a^{m+n}[/latex]
2. Second Law
[latex]\ a^{m}\div a^{n}=a^{m-n} , \qquad a\neq0[/latex]
3. Third Law
[latex] \left(a^{m}\right)^{n}=a^{mn}[/latex]
4. Zero Index Law
[latex]a^{0}=1 , \qquad a\neq0[/latex]
5. Negative Index Law
[latex]a^{-n}=\dfrac{1}{a^{n}} , \qquad a\neq0[/latex]
6. Fractional Index Law
[latex]a^{1/n}=\sqrt[n]{a}[/latex]
These laws should be memorised.
