An index in mathematics, also known as a power or exponent, is a small number or variable placed above and to the right of a base number or variable. It indicates how many times the base number should be multiplied by itself. The plural form of index is indices.
Understanding Indices
At its core, an index simplifies the representation of repeated multiplication. Instead of writing $2 \times 2 \times 2 \times 2$, we can simply write $2^4$, where '2' is the base and '4' is the index (or exponent). This means the base (2) is multiplied by itself 4 times. For instance, in an expression like $2^4$, the number 4 is the index of 2, indicating $2 \times 2 \times 2 \times 2 = 16$.
Indices are fundamental in algebra, applying to both constants (values that do not change, like 5 in $5^2$) and variables (symbols representing unknown values, like x in $x^3$).
Key Components of an Indexed Expression
Every expression involving an index consists of two main parts:
- Base: The number or variable that is being multiplied.
- Index (or Exponent/Power): The small number or variable that tells you how many times to multiply the base by itself.
| Component | Description | Example Expression | Example Component |
|---|---|---|---|
| Base | The number or variable being multiplied. | $5^3$ | 5 |
| Index | The number of times the base is multiplied. | $5^3$ | 3 |
Why Are Indices Important?
Indices provide a concise and efficient way to write very large or very small numbers, making calculations and algebraic manipulations much simpler. They are crucial in various fields, from science and engineering to finance and computer science.
Types of Indices
Indices can take several forms, each with a specific meaning:
-
Positive Indices
A positive integer index indicates straightforward repeated multiplication.
- Example: $3^2 = 3 \times 3 = 9$ (3 to the power of 2)
- Example: $x^5 = x \times x \times x \times x \times x$
-
Zero Index
Any non-zero base raised to the power of zero is always 1.
- Rule: $a^0 = 1$ (where $a \ne 0$)
- Example: $7^0 = 1$
- Example: $(y+z)^0 = 1$
-
Negative Indices
A negative index indicates the reciprocal of the base raised to the positive value of that index.
- Rule: $a^{-n} = \frac{1}{a^n}$ (where $a \ne 0$)
- Example: $5^{-2} = \frac{1}{5^2} = \frac{1}{25}$
- Example: $x^{-3} = \frac{1}{x^3}$
-
Fractional Indices
A fractional index, such as $\frac{1}{n}$, indicates a root of the base. If the index is $\frac{m}{n}$, it means the $n$-th root of the base raised to the power of $m$.
- Rule: $a^{1/n} = \sqrt[n]{a}$ (the n-th root of a)
- Rule: $a^{m/n} = (\sqrt[n]{a})^m = \sqrt[n]{a^m}$
- Example: $9^{1/2} = \sqrt{9} = 3$
- Example: $8^{2/3} = (\sqrt[3]{8})^2 = 2^2 = 4$
Laws of Indices
To perform operations with indices efficiently, several rules or "laws of indices" are applied:
-
Multiplication Rule: When multiplying powers with the same base, add the indices.
- $a^m \times a^n = a^{m+n}$
- Example: $2^3 \times 2^4 = 2^{3+4} = 2^7$
-
Division Rule: When dividing powers with the same base, subtract the indices.
- $a^m \div a^n = a^{m-n}$
- Example: $5^6 \div 5^2 = 5^{6-2} = 5^4$
-
Power of a Power Rule: When raising a power to another power, multiply the indices.
- $(a^m)^n = a^{m \times n}$
- Example: $(3^2)^3 = 3^{2 \times 3} = 3^6$
-
Power of a Product Rule: The power of a product is the product of the powers.
- $(ab)^n = a^n b^n$
- Example: $(2x)^3 = 2^3 x^3 = 8x^3$
-
Power of a Quotient Rule: The power of a quotient is the quotient of the powers.
- $(\frac{a}{b})^n = \frac{a^n}{b^n}$
- Example: $(\frac{x}{y})^4 = \frac{x^4}{y^4}$
These laws simplify complex expressions and are fundamental to solving equations involving exponents. For more detailed information on specific topics like exponents, you can refer to resources like Wikipedia's page on Exponentiation or Khan Academy's lessons on exponents.
