What Are the Laws of Indices Definition Rules and Solved Examples
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.
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
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:
\[\frac{5^{6}}{5^{4}}\] = \[5^{6-4}\]
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:
\[(2^{3})^{4}\] = \[(2^{12})\]
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:
\[(3^{-2})\] = \[\frac{1}{3^{2}}\] = \[\frac{1}{9}\]
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
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:
\[7^{2}\] . \[5^{2}\] = \[35^{2}\]
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:
\[4^{2}\] . \[2^{4}\] = \[2^{4}\]
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.
Multiplication
\[log_{b}\] (x . y) = \[log_{b}\] (x) + \[log_{b}\] (y)
Example:
\[log_{10}\] (2 . 3) =\[log_{10}\] (2) + \[log_{10}\] (3)
Division
\[log_{b}\] \[\frac{x}{y}\] = \[log_{b}\] (x) - \[log_{b}\] (y)
Example:
\[log_{10}\] \[\frac{2}{3}\] = \[log_{10}\] (2) - \[log_{10}\] (3)
Power of Power
\[log_{b}\] \[x^{m}\] = m. \[log_{b}\] (x)
Example:
\[log_{10}\] \[4^{2}\] = 2 \[log_{10}\] (4)
Zero Power
\[log_{b}\] 1 = 0
1 = \[b^{x}\] , then x=0.
Negative power
\[log_{b}\] \[\frac{1}{x}\] = - \[log_{b}\] (x)
Example:
\[log_{10}\] \[\frac{1}{2}\] = - \[log_{10}\] (2)
Singular Index
\[log_{b}\] = 1
Example:
\[log_{10}\] = 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)
FAQs on Laws of Indices Explained with Rules and Examples
1. What are the laws of indices?
The laws of indices are rules that simplify expressions involving powers or exponents. The main exponent rules are:
- am × an = am+n (product rule)
- am ÷ an = am−n (quotient rule)
- (am)n = amn (power of a power)
- (ab)n = anbn (power of a product)
- a0 = 1 (zero index rule, a ≠ 0)
- a−n = 1/an (negative index rule)
2. What is the product rule of indices?
The product rule of indices states that when multiplying powers with the same base, you add the exponents: am × an = am+n.
- Example: 23 × 24 = 23+4 = 27 = 128
- This rule only works when the base is the same.
3. What is the quotient rule of indices?
The quotient rule of indices states that when dividing powers with the same base, you subtract the exponents: am ÷ an = am−n (a ≠ 0).
- Example: 56 ÷ 52 = 56−2 = 54 = 625
- Ensure the bases are identical before applying the rule.
4. What happens when a power is raised to another power?
When a power is raised to another power, you multiply the exponents: (am)n = amn.
- Example: (32)4 = 32×4 = 38 = 6561
- This is called the power of a power rule.
5. What is the zero index rule?
The zero index rule states that any non-zero number raised to the power of zero equals 1, that is a0 = 1 (a ≠ 0).
- Example: 70 = 1
- This rule follows from the quotient rule: am ÷ am = a0 = 1.
6. What is a negative index?
A negative index means you take the reciprocal of the base: a−n = 1/an (a ≠ 0).
- Example: 2−3 = 1/23 = 1/8
- It does not make the number negative; it makes it a fraction.
7. How do you simplify expressions using the laws of indices?
To simplify expressions using the laws of indices, apply the appropriate exponent rules step by step.
- Example: (23 × 24) ÷ 25
- Step 1: Add exponents → 27
- Step 2: Subtract exponents → 27−5 = 22
- Final answer: 4
8. What is the power of a product rule?
The power of a product rule states that (ab)n = anbn.
- Example: (2 × 3)2 = 62 = 36
- Using the rule: 22 × 32 = 4 × 9 = 36
9. What is the difference between positive and negative indices?
The difference between positive and negative indices is that positive exponents represent repeated multiplication, while negative exponents represent reciprocals.
- Positive index: 32 = 3 × 3 = 9
- Negative index: 3−2 = 1/32 = 1/9
10. What are common mistakes when using the laws of indices?
Common mistakes in using the laws of indices include applying rules incorrectly or ignoring base conditions.
- Adding exponents when bases are different (incorrect: 23 × 32)
- Forgetting that a0 = 1 (a ≠ 0)
- Thinking a−n makes the number negative
- Multiplying exponents instead of adding in the product rule
