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Laws of Indices

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Last updated date: 17th Apr 2024
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What is Indices?

Index is referred to as the power or exponent raised to a number or variable. Index in its plural form is termed as indices. If we write 2³ or a⁵, here 3 and 5 and indices. Each number naturally has an index of 1 but we do not write it as it does not denote any change of value mathematically. If the index is anything other than 1, we require to write it down as the power of the base number. The index of a number can also be zero or negative.

The index represents the number of times a number has to be multiplied by itself. These numbers are governed by several indices rules that we will discuss here. Given below is the representation of the index of a number.

\[a^{n}\] = \[a \ast a \ast a \ast a\cdot \cdot \cdot \cdot\] (n times)

Here,

a  is the base and n is termed as the index.

\[2^{4}\] = \[2 \ast 2 \ast 2 \ast 2\] = 16

\[10^{3}\] = \[10 \ast 10 \ast 10\] = 1000

As per indices definition, a number or a variable may have an index. It tells us about how many times the base number is to be multiplied by itself.


Theory of Indices

The laws of indices are a set of fundamental rules that govern the way indexes or indices are to be dealt with mathematically. Indices are not just used to improve the ease of writing the numbers mathematically but also have a specific function and therefore these indices rules are of utmost importance.

Only after knowing these Laws of Indices rules can you solve the algebraic indices problems

We will look at each law of indices formula with index laws examples one by one for various algebraic indices.


Laws of Indices Formulas

Given below are all the laws of indices that you will encounter while dealing with indices. No matter how complex the problem is, these are all the fundamental laws that govern the indices rules.

  1. Multiplication

If two terms with a similar base are to be multiplied by each other, the indices have to be added.

aⁿ . aᵐ = aⁿ⁺ᵐ

Example:

4³ . 4⁶ = 4³⁺⁶ = 4⁹

  1. Division

If two terms with a similar base are to be divided, the indices have to be subtracted

\[\frac{a^{n}}{a^{m}}\] = \[a^{n-m}\]

Example:

\[\frac{5^{6}}{5^{4}}\] = \[5^{6-4}\]

  1. Power of a Power

If the index of a number is itself raised into another power, then the two indices have to be multiplied.

\[(a^{n})^{m}\] = \[(a^{nm})\]

Example:

\[(2^{3})^{4}\] = \[(2^{12})\]

  1. Negative Power

If a term has a negative index it can be represented as reciprocal with the positive index as its power.

\[(a^{-n})\] = \[\frac{1}{a^{n}}\]

Example:

\[(3^{-2})\] = \[\frac{1}{3^{2}}\] = \[\frac{1}{9}\]

  1. Zero Power

If a term has the index as 0, then the value of the term becomes one, no matter what the base value is.

\[a^{0}\] = 1

Example:

\[5^{0}\] = 1

  1. Multiplication with Similar Indices and Different Base

If two terms in multiplication with each other have similar indices but different bases, then the two bases are multiplied with each other.

\[a^{n}\] . \[b^{n}\] = \[(ab)^{n}\]

Example:

\[7^{2}\] . \[5^{2}\] = \[35^{2}\]

  1. Division with Similar Indices and Different Base

If two terms in a division with each other have similar indices but different bases, then the two bases are to be divided with each other.

\[\frac{a^{n}}{b^{n}}\] = \[\left (\frac{a}{b}  \right )^{n}\]

Example:

\[4^{2}\] . \[2^{4}\] = \[2^{4}\]

  1. Fractional index

If a term has index in the fraction form it can be represented in the radical form as well.

\[a^{\frac{n}{m}}\] = \[\left ( \sqrt[m]{a} \right )^{n}\]

Example:

\[4^{\frac{2}{3}}\] = \[\left ( \sqrt[3]{4} \right )^{2}\]

You can download the law of indices pdf to revise these index laws examples from time to time in order to be fluent with them.


Laws of Logarithms

Using the Indices rules, we can formulate the laws of indices and logarithms.

  1. Multiplication

\[log_{b}\] (x . y) = \[log_{b}\] (x) + \[log_{b}\] (y)

Example:

\[log_{10}\] (2 . 3) =\[log_{10}\] (2) + \[log_{10}\] (3) 

  1. Division

\[log_{b}\] \[\frac{x}{y}\] = \[log_{b}\] (x) - \[log_{b}\] (y)

Example:

\[log_{10}\] \[\frac{2}{3}\] = \[log_{10}\] (2) - \[log_{10}\] (3)

  1. Power of Power

\[log_{b}\] \[x^{m}\] = m. \[log_{b}\] (x)

Example:

\[log_{10}\] \[4^{2}\] = 2  \[log_{10}\] (4)

  1. Zero Power

\[log_{b}\] 1 = 0

1 = \[b^{x}\] , then x=0.

  1. Negative power

\[log_{b}\] \[\frac{1}{x}\] = - \[log_{b}\] (x)

Example:

\[log_{10}\] \[\frac{1}{2}\] = - \[log_{10}\] (2)

  1. Singular Index

\[log_{b}\] = 1

Example:

\[log_{10}\] = 1

  1. Fractional Power

\[log_{b}\] \[\left ( \sqrt[n]{x} \right )\] = \[\left ( \frac{1}{n} \right )\] \[log_{b}\] (x)

Example:

\[log_{10}\] \[\left ( \sqrt[3]{5} \right )\] = \[\left ( \frac{1}{3} \right )\] \[log_{10}\] (5)

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FAQs on Laws of Indices

1. Comment on laws of indices and logarithms?

Logarithms are the mirror images of the index. These are written as:

X = \[a^{n}\]

\[log_{a}\] x = n

We read this as the log of x to base a is n.

Y = \[x^{n}\]

\[log_{x}\] y = n The log of y to the base x is n.

All the logs must be in the same base to apply the rules of logs. Base 10 and base e is the most common base you will encounter while solving problems. Logs to the base e are termed natural logarithms.

\[log_{e}\] y = ln y

Y = exp(x) = \[e^{x}\]

\[log_{e}\] y = x

Ln y = x

2. Explain the properties of indices?

The properties of indices can also be called the law of indices formula and it gives us the fundamental rules that are used in the index problems.

The properties of indices are:

Multiplication Property:

\[a^{n}\] . \[a^{m}\] = \[a^{n+m}\]

Division Property:

\[\frac{a^{n}}{a^{m}}\] = \[a^{n-m}\]

Power of a Power:

\[\left ( a^{n} \right )^{m}\] = \[a^{nm}\]

Negative Power:

\[a^{-n}\] = \[\frac{1}{a^{n}}\]

Zero Power:

\[a^{0}\] = 1

Multiplication with Similar Indices Different Bases:

\[a^{n}\] . \[b^{n}\] = \[\left ( ab \right )^{n}\]

Division with similar Indices Different Bases:

\[\frac{a^{n}}{b^{n}}\] = \[\left (\frac{a}{b}  \right )^{n}\]

Fractional Index:

\[a^{\frac{n}{m}}\] = \[\left ( \sqrt[m]{a} \right )^{n}\]