Table of Contents
Understanding Exponents
In this chapter we look closely at what exponents are, how to read and write them, and how to work with very simple exponential expressions. General “laws of exponents” and roots will be treated in nearby chapters; here we focus on the basic idea and the most direct uses.
What an Exponent Means
An exponent tells you how many times to multiply a number by itself.
If a number $a$ is written with an exponent $n$ like this:
$$a^n$$
then:
- $a$ is called the base.
- $n$ is called the exponent (or power).
- $a^n$ means “$a$ multiplied by itself $n$ times.”
Formally, for a positive whole number $n$:
- $a^1 = a$
- $a^2 = a \cdot a$
- $a^3 = a \cdot a \cdot a$
- In general:
$$a^n = \underbrace{a \cdot a \cdot \dots \cdot a}_{n\ \text{factors of }a}$$
Example:
- $2^4 = 2 \cdot 2 \cdot 2 \cdot 2 = 16$
- $5^3 = 5 \cdot 5 \cdot 5 = 125$
Notice that the exponent counts how many copies of the base appear in the multiplication.
Reading and Naming Common Powers
Some exponents have special spoken names:
- $a^2$ is read “$a$ squared”.
- $a^3$ is read “$a$ cubed”.
- $a^4$ is read “$a$ to the fourth power”.
- In general, $a^n$ is read “$a$ to the $n$th power”.
Examples:
- $3^2$ is “three squared”.
- $4^3$ is “four cubed”.
- $10^5$ is “ten to the fifth power”.
These names are especially common when the base is a number.
Expanding and Evaluating Simple Powers
To expand a power means to write out the repeated multiplication. To evaluate means to find its numerical value.
Example 1:
- Expand $3^4$:
$^4 = 3 \cdot 3 \cdot 3 \cdot 3$$ - Evaluate:
$ \cdot 3 = 9,\quad 9 \cdot 3 = 27,\quad 27 \cdot 3 = 81$$
So ^4 = 81$.
Example 2:
- Expand $(-2)^3$:
$$(-2)^3 = (-2)\cdot(-2)\cdot(-2)$$ - Multiply:
$$(-2)\cdot(-2) = 4,\quad 4\cdot(-2) = -8$$
So $(-2)^3 = -8$.
Be careful with negative signs:
- $(-2)^3$ means the base is $-2$ and exponent is $3$.
- $-2^3$ means “negative of $2^3$”:
$$-2^3 = -(2^3) = -8$$
In $(-2)^3$, the negative is inside the base; in $-2^3$, it is outside and applied after the exponent.
Zero and One as Exponents
Two special exponents that appear very often are $0$ and $1$.
- Exponent $1$: multiplying a number by itself once is just the number:
$$a^1 = a$$
for any number $a$. - Exponent $0$: for any nonzero number $a$,
$$a^0 = 1$$
Examples:
- $7^1 = 7$
- $(-5)^1 = -5$
- $2^0 = 1$
- $10^0 = 1$
- $(-3)^0 = 1$
You should not use $0^0$ in basic arithmetic; it is not given a consistent value in simple settings and is usually left undefined.
Even and Odd Exponents with Negative Bases
When the base is negative and the exponent is a positive whole number, the sign of the result depends on whether the exponent is even or odd.
Let the base be $-a$ (with $a>0$):
- If the exponent is even:
$$(-a)^{2k} \text{ is positive}$$ - If the exponent is odd:
$$(-a)^{2k+1} \text{ is negative}$$
Examples:
- $(-3)^2 = (-3)\cdot(-3) = 9$ (even exponent, result positive).
- $(-3)^3 = (-3)\cdot(-3)\cdot(-3) = -27$ (odd exponent, result negative).
- $(-1)^4 = 1$; $(-1)^5 = -1$.
This is a quick way to determine the sign of a power with a negative base without doing all the multiplications.
Exponents with Base 10
Powers of 10 are especially useful and appear everywhere in arithmetic and science.
For positive whole numbers $n$:
$$10^n = 1\text{ followed by }n\text{ zeros}.$$
Examples:
- $10^1 = 10$
- $10^2 = 100$
- $10^3 = 1000$
- $10^4 = 10000$
You can use this to rewrite numbers:
- $3000 = 3 \cdot 1000 = 3 \cdot 10^3$
- $50\,000 = 5 \cdot 10^4$
For negative exponents (which are treated more fully elsewhere), the pattern continues in the other direction, but in this chapter you only need to be comfortable with positive exponents and seeing large numbers as multiples of powers of 10.
Exponents in Words and Repeated Multiplication
You should be able to move between:
- Words,
- Repeated multiplication,
- Exponential notation.
Examples:
- “Four times four times four times four”
Repeated multiplication: \cdot4\cdot4\cdot4$
Exponential form: ^4$ - “Seven squared”
Repeated multiplication: \cdot7$
Exponential form: ^2$ - $2^5$
Words: “two to the fifth power” or “two to the power of five”
Repeated multiplication: \cdot2\cdot2\cdot2\cdot2$
Practice going in both directions to become fluent with exponents.
Simple Exponents in Expressions
Exponents often appear inside larger expressions. You need to evaluate the powers before doing addition or subtraction, according to the usual order of operations.
Examples:
- $$3^2 + 4 = ?$$
Evaluate the exponent first:
$^2 = 9,$$
so:
$^2 + 4 = 9 + 4 = 13.$$ - $$2\cdot 5^2 = ?$$
First compute ^2 = 25$, then multiply:
$\cdot 25 = 50.$$ - $$6 - (-2)^2 = ?$$
Compute $(-2)^2 = 4$, then subtract:
$ - 4 = 2.$$
Remember: in an expression, exponents are evaluated before multiplication and division, which come before addition and subtraction.
Common Mistakes to Avoid
- Forgetting parentheses with negative numbers
- Correct: $(-3)^2 = 9$
- Without parentheses: $-3^2 = -(3^2) = -9$
If the whole negative number is the base, enclose it in parentheses.
- Thinking $a^n$ means $na$ (multiplying by the exponent)
- $3^4$ is $3\cdot3\cdot3\cdot3 = 81$, not $3\cdot4 = 12$.
- Exponent means repeated multiplication, not repeated addition.
- Applying the exponent to only part of the base
The exponent applies to whatever it is directly attached to:
- $2\cdot3^2$ means $2\cdot(3^2)$, not $(2\cdot3)^2$.
- $(2\cdot3)^2 = 6^2 = 36$, but $2\cdot3^2 = 2\cdot9 = 18$.
Careful reading of the notation is essential.
Practice-Style Prompts (No Solutions Here)
To strengthen your understanding, try exercises of these types:
- Rewrite using exponents:
- $5\cdot5\cdot5\cdot5$
- $(-2)\cdot(-2)\cdot(-2)$
- Expand and evaluate:
- $4^3$
- $(-3)^4$
- $10^2$
- Determine the sign (positive or negative) without full calculation:
- $(-2)^7$
- $(-5)^6$
- Evaluate expressions with exponents:
- $2^3 + 5$
- $3\cdot 4^2$
- $7 - (-1)^3$
These activities help you become comfortable reading, writing, and evaluating exponents, preparing you for the more general rules and connections with roots in the following chapters.