Power: The Language of Repeated Multiplication
The Foundation: Base, Exponent, and Power
Let's start with the basics. A power is composed of two main parts:
- The Base: This is the number that is being multiplied by itself.
- The Exponent (or Index): This is the small number written to the upper right of the base. It tells you how many times to use the base as a factor in the multiplication.
The general form is written as $a^n$, where $a$ is the base and $n$ is the exponent. We say this as "a to the power of n" or more simply "a to the n."
$5^3 = 5 \times 5 \times 5 = 125$.
Here, $5$ is the base, $3$ is the exponent, and $125$ is the power (or the value of the expression).
This notation is incredibly powerful because it saves a tremendous amount of time and space. Imagine writing out a number like $10 \times 10 \times 10 \times 10 \times 10$. With exponents, we can simply write $10^5$.
Special Exponents You Must Know
Certain exponents have special names and rules that make calculations easier.
| Exponent | Name | Rule | Example |
|---|---|---|---|
| $a^0$ | Zero Exponent | Any non-zero number to the power of zero is 1. | $7^0 = 1$, $(-123)^0 = 1$ |
| $a^1$ | First Power | Any number to the power of 1 is the number itself. | $9^1 = 9$ |
| $a^{-n}$ | Negative Exponent | A negative exponent means "one over" the positive power. | $4^{-2} = \frac{1}{4^2} = \frac{1}{16}$ |
| $a^{1/n}$ | Fractional Exponent (Root) | An exponent of 1/n is the n-th root of the base. | $8^{1/3} = \sqrt[3]{8} = 2$ |
The Essential Laws of Exponents
To work confidently with powers, you need to master a few key rules, often called the "Laws of Exponents." These laws allow you to simplify expressions and solve equations involving powers.
When multiplying two powers with the same base, you add the exponents.
Formula: $a^m \times a^n = a^{m+n}$
Example: $2^3 \times 2^4 = (2 \times 2 \times 2) \times (2 \times 2 \times 2 \times 2) = 2^{3+4} = 2^7 = 128$
When dividing two powers with the same base, you subtract the exponents.
Formula: $\frac{a^m}{a^n} = a^{m-n}$ (where $a \neq 0$)
Example: $\frac{5^6}{5^2} = 5^{6-2} = 5^4 = 625$
To raise a power to another power, you multiply the exponents.
Formula: $(a^m)^n = a^{m \times n}$
Example: $(3^2)^4 = 3^{2 \times 4} = 3^8 = 6561$
When a product is raised to a power, each factor is raised to that power.
Formula: $(ab)^n = a^n \times b^n$
Example: $(2 \times 5)^3 = 2^3 \times 5^3 = 8 \times 125 = 1000$
When a quotient is raised to a power, both the numerator and denominator are raised to that power.
Formula: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ (where $b \neq 0$)
Example: $\left(\frac{3}{4}\right)^2 = \frac{3^2}{4^2} = \frac{9}{16}$
Powers in Action: Real-World Applications
Powers are not just abstract mathematical concepts; they are used to describe phenomena in science, finance, and computer science.
Scientific Notation: This is a way to express very large or very small numbers concisely. A number in scientific notation is written as a product of a number between 1 and 10 and a power of 10. For example, the speed of light is approximately $300,000,000$ meters per second. This is cumbersome to write, so we use powers: $3 \times 10^8$ m/s. Similarly, the mass of a dust particle is about $0.000000000753$ kg, which is $7.53 \times 10^{-10}$ kg.
Exponential Growth: This occurs when a quantity increases by a fixed percentage over equal time periods. It is modeled by functions with a variable in the exponent, like $A = P(1 + r)^t$ for compound interest. If you invest $P=100$ dollars at an interest rate of $r=0.05$ (5%) per year, the amount after $t=10$ years is $A = 100(1+0.05)^{10} = 100(1.05)^{10} \approx 162.89$ dollars. The power here captures the compounding effect, where you earn interest on your previous interest.
Computer Science: In the digital world, everything is represented using binary digits (bits), which are 0s and 1s. The number of unique values that can be represented by $n$ bits is $2^n$. For example, a single byte is 8 bits, so it can represent $2^8 = 256$ different values. This principle is fundamental to how computers store and process information.
Common Mistakes and Important Questions
Q: Is the exponent the same as the power?
No, this is a common point of confusion. The exponent is the small number that indicates how many times to multiply the base. The power is the entire expression (e.g., $2^5$) and can also refer to the final result or value of that expression (e.g., $32$). In the expression $2^5$, $5$ is the exponent and $2^5$ is the "fifth power of 2".
Q: What is the result of a negative base with an exponent?
The result depends on whether the exponent is even or odd.
Even Exponent: $(-2)^4 = (-2) \times (-2) \times (-2) \times (-2) = 16$ (positive result).
Odd Exponent: $(-2)^3 = (-2) \times (-2) \times (-2) = -8$ (negative result).
Be very careful with parentheses: $-2^4$ means $-(2^4) = -16$, which is different from $(-2)^4 = 16$.
Q: Why is any number to the power of zero equal to one?
This can be understood using the Quotient of Powers rule. Consider $\frac{a^3}{a^3}$. We know this equals 1 because any number divided by itself is 1. Using the exponent rule, we also have $\frac{a^3}{a^3} = a^{3-3} = a^0$. For this to be consistent, $a^0$ must equal 1. This holds for any non-zero base $a$.
Footnote
This article uses the terms exponent and index interchangeably. In some regions, "index" is the preferred term, while "exponent" is more common in North America. Both refer to the number $n$ in the expression $a^n$.