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.
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
1. What is an index in mathematics and what does it represent?
In mathematics, an index (plural: indices) refers to the power or exponent to which a number or variable, known as the base, is raised. It indicates how many times the base is to be multiplied by itself. For example, in the expression 5³, the number 5 is the base and 3 is the index, which means 5 is multiplied by itself three times (5 × 5 × 5 = 125).
2. What are the main laws of indices used in algebra?
The main laws of indices are a set of rules that simplify expressions involving powers. The most fundamental laws are:
- Multiplication Law: To multiply terms with the same base, add their indices (aⁿ ⋅ aᵐ = aⁿ⁺ᵐ).
- Division Law: To divide terms with the same base, subtract the index of the denominator from the numerator (aⁿ / aᵐ = aⁿ⁻ᵐ).
- Power of a Power Law: To raise a power to another power, multiply the indices ((aⁿ)ᵐ = aⁿᵐ).
- Zero Index Law: Any non-zero base raised to the power of zero is equal to 1 (a⁰ = 1).
- Negative Index Law: A base raised to a negative power is the reciprocal of the base raised to the positive power (a⁻ⁿ = 1/aⁿ).
- Fractional Index Law: A fractional index represents a root, where a^(n/m) is the m-th root of 'a' raised to the power 'n'.
3. How do the rules for zero and negative indices work?
The rules for zero and negative indices provide a way to handle exponents that are not positive integers.
- The Zero Index Rule (a⁰ = 1) states that any non-zero number or variable raised to the power of zero equals 1. This is a crucial definition that maintains consistency in other index laws. For instance, 7⁰ = 1 and (x+y)⁰ = 1.
- The Negative Index Rule (a⁻ⁿ = 1/aⁿ) states that a term with a negative exponent is equivalent to its reciprocal with a positive exponent. For example, 3⁻² is the same as 1/3², which equals 1/9. This rule is essential for simplifying algebraic fractions.
4. What is the difference between (aⁿ)ᵐ and aⁿ ⋅ aᵐ?
This highlights a common area of confusion. The key difference lies in the operation being performed on the indices.
- (aⁿ)ᵐ follows the 'Power of a Power' rule. Here, an expression that is already a power (aⁿ) is being raised to another power (m). In this case, you must multiply the indices: (aⁿ)ᵐ = aⁿᵐ. For example, (2³)⁴ = 2³ˣ⁴ = 2¹².
- aⁿ ⋅ aᵐ follows the 'Multiplication' rule. Here, two terms with the same base ('a') are being multiplied. In this case, you must add the indices: aⁿ ⋅ aᵐ = aⁿ⁺ᵐ. For example, 2³ ⋅ 2⁴ = 2³⁺⁴ = 2⁷.
5. How does the law for fractional indices relate to roots and powers?
The law for fractional indices provides a direct link between exponents and radicals (roots). An expression like a^(n/m) can be interpreted in two parts: the denominator 'm' represents the root, and the numerator 'n' represents the power. Therefore, a^(n/m) means 'take the m-th root of 'a', and then raise the result to the power of n'. For example, to solve 8^(2/3), you would first find the cube root of 8 (which is 2) and then square the result, giving 2² = 4.
6. Why is it essential for the bases to be the same when applying the multiplication or division laws of indices?
It is essential for the bases to be the same because an index represents repeated multiplication of its specific base. The multiplication law (aⁿ ⋅ aᵐ = aⁿ⁺ᵐ) is a shortcut for counting how many times the base 'a' is multiplied by itself in total. For example, 2³ ⋅ 2² means (2×2×2) × (2×2), which is a total of five 2s multiplied together, or 2⁵. If the bases are different, such as in 2³ ⋅ 5², you have (2×2×2) × (5×5). Since there is no common base, you cannot combine the exponents by adding them. The expression can only be simplified by calculating the values separately (8 × 25 = 200).
7. How are the laws of indices foundational for understanding logarithms?
The laws of indices are foundational for logarithms because a logarithm is the inverse operation of exponentiation. Each law of indices has a corresponding law of logarithms. For example:
- The index law aˣ ⋅ aʸ = aˣ⁺ʸ (when you multiply numbers, you add their powers) corresponds to the logarithm law log(m ⋅ n) = log(m) + log(n).
- The index law (aˣ)ʸ = aˣʸ (for a power of a power, you multiply the powers) corresponds to the logarithm law log(mⁿ) = n ⋅ log(m).
