Index Notation: The Power of Shortening Numbers
The Foundation: Base, Exponent, and Power
Let's start with the basic building blocks. In index notation, an expression is written as $a^n$.
- Base ($a$): This is the number that is being multiplied by itself.
- Exponent or Index ($n$): This is the small number written to the upper right of the base. It indicates how many times the base is used as a factor.
- Power: The entire expression $a^n$ is called a power. We can say "$a$ to the power of $n$" or "$a$ to the $n$-th power." The result of this multiplication is also called the value of the power.
Example: $5^3$ means $5 \times 5 \times 5$. Here, $5$ is the base, $3$ is the exponent, and the value of the power is $125$.
Special Exponents and Their Rules
What happens when the exponent is not a simple positive number? Index notation has elegant answers for these special cases, governed by clear laws or "rules of exponents."
| Rule Name | Expression | Example | Explanation |
|---|---|---|---|
| Zero Exponent | $a^0 = 1$ (for $a \neq 0$) | $7^0 = 1$, $(-3)^0 = 1$ | Any non-zero number raised to the power of zero equals 1. |
| Negative Exponent | $a^{-n} = \frac{1}{a^n}$ | $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$ | A negative exponent means "take the reciprocal[3] of the base raised to the positive exponent." |
| Product of Powers | $a^m \times a^n = a^{m+n}$ | $x^4 \times x^2 = x^{4+2} = x^6$ | When multiplying powers with the same base, add the exponents. |
| Quotient of Powers | $\frac{a^m}{a^n} = a^{m-n}$ | $\frac{5^7}{5^4} = 5^{7-4} = 5^3$ | When dividing powers with the same base, subtract the exponents. |
| Power of a Power | $(a^m)^n = a^{m \times n}$ | $(y^2)^3 = y^{2 \times 3} = y^6$ | When raising a power to another power, multiply the exponents. |
These rules are not random; they follow logically from the definition of index notation. For instance, the Zero Exponent rule can be derived from the Quotient of Powers rule: $\frac{a^n}{a^n} = a^{n-n} = a^0$, and we know that any number divided by itself equals 1 (as long as it's not zero).
Index Notation in the Real World: Scientific Notation
One of the most important applications of index notation is scientific notation. Scientists, engineers, and mathematicians use it to conveniently write numbers that are extremely large or incredibly small. This is common in astronomy, physics, chemistry, and biology.
A number is written in scientific notation when it is in the form $c \times 10^n$, where $1 \leq |c| < 10$ (this is called the coefficient) and $n$ is an integer (the exponent).
| Standard Form | Scientific Notation | Real-World Context |
|---|---|---|
| 149,600,000 km | $1.496 \times 10^8$ km | Average distance from Earth to the Sun (1 Astronomical Unit[4]). |
| 0.0000000001 m | $1.0 \times 10^{-10}$ m | Approximate diameter of a hydrogen atom. |
| 6,022,000,000,000,000,000,000 | $6.022 \times 10^{23}$ | Avogadro's constant (number of particles in one mole). |
Performing calculations with numbers in scientific notation is easier because you can use the rules of exponents. To multiply $(3 \times 10^4) \times (2 \times 10^5)$, multiply the coefficients ($3 \times 2 = 6$) and add the exponents ($10^4 \times 10^5 = 10^{9}$), giving $6 \times 10^9$.
Important Questions
Q1: What is the difference between a base and an exponent?
Q2: Why does any number (except zero) to the power of zero equal 1?
Q3: How do you handle exponents with different bases? For example, can you simplify $2^3 \times 3^2$?
Footnote
[1] Scientific Notation: A method of writing numbers as a product of a number between 1 and 10 and a power of 10. Used for very large or very small numbers.
[2] Exponential Growth: A process where the rate of growth of a quantity is proportional to its current value, leading to growth that becomes ever more rapid. It is modeled by functions of the form $y = a \cdot b^x$ where $b > 1$.
[3] Reciprocal: The multiplicative inverse of a number. For a number $a$, its reciprocal is $\frac{1}{a}$ (provided $a \neq 0$).
[4] Astronomical Unit (AU): A standard unit of measurement in astronomy, approximately equal to the mean distance from the center of the Earth to the center of the Sun.