Exponents provide a compact way to represent repeated multiplication and to simplify calculations involving very large and very small quantities.
Exponent
The small raised number in a power, for example the $n$ in $10^{n}$, which tells how many times the base is multiplied by itself. Negative exponents represent reciprocals.
Base
The number being raised to a power in an exponential expression.
Power
A number being multiplied by itself a specific number of times. Power is the whole expression including the base and the exponent.
Exponents Represent Repeated Multiplication (And Sign Matters)
- When $n$ is a positive integer, $a^n$ means multiply $a$ by itself $n$ times: $$a^n=\underbrace{a\times a\times\cdots\times a}_{n\text{ factors}}$$
- For example:
- $5^3=5\times 5\times 5=125$
- $(-2)^5=-2\times -2\times -2\times -2\times -2=-32$
- $(-7)^2=(-7)\times(-7)=49$
- A key idea is the difference between a negative base and a negative exponent.
- They affect the result in completely different ways.
A common mistake is confusing $-2^4$ with $(-2)^4$.
- $-2^4=-(2^4)=-16$ (the exponent applies to 2 only)
- $(-2)^4=16$ (the exponent applies to the negative base)
Use parentheses whenever the base is negative.
Exponent Laws Let You Simplify Quickly
- Exponent laws (also called laws of indices) allow you to simplify expressions without expanding them fully.
- These are especially useful in algebra and in scientific notation.
- Throughout this section, assume the bases are real numbers and watch for restrictions like division by zero.
Product Rule: Same Base, Add Exponents
- If the base is the same, multiplying powers adds exponents: $$a^m\times a^n=a^{m+n}$$
- Reason: you are combining $m$ factors of $a$ and $n$ factors of $a$ into $m+n$ factors.
$$4^3\times 4^8=4^{3+8}=4^{11}$$
The product rule only works when the base is the same. For example, $3^5\times 2^5$ does not become $5^?$. Instead you can use "power of a product" (see below) because the exponents match: $3^5\times 2^5=(3\cdot 2)^5=6^5$.
Power Of A Product: Distribute The Exponent
When a product is raised to a power, each factor gets the power: $$(ab)^n=a^n b^n$$.
$$(2\cdot 3)^4=2^4\cdot 3^4$$
This is also the key idea behind simplifying expressions like $3^5\times(-2)^5$: $$3^5\times(-2)^5=(3\cdot -2)^5=(-6)^5$$
Quotient Rule: Same Base, Subtract Exponents
When dividing powers with the same base: $$\frac{a^m}{a^n}=a^{m-n} \quad (a\ne 0)$$
$$\frac{12^{15}}{12^2}=12^{15-2}=12^{13}$$
Power Of A Quotient: Distribute The Exponent Over Division
$$\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n} \quad (b\ne 0)$$
This matches the textbook form: $$\frac{a^n}{b^n}=\left(\frac{a}{b}\right)^n$$
$$\frac{(-48)^7}{4^7}=\left(\frac{-48}{4}\right)^7=(-12)^7$$
Power Rule (Power Of A Power): Multiply Exponents
$$\left(a^m\right)^n=a^{mn}$$
$$\left(2^4\right)^3=2^{4\cdot 3}=2^{12}$$
- Think of $\left(a^m\right)^n$ as "$n$ groups of $m$ factors of $a$".
- That makes $mn$ total factors of $a$, which is why the exponents multiply.
Zero And Negative Exponents Give Meaningful Extensions
Exponent laws are most powerful when we extend them beyond positive integers.
Zero Exponent Means "Equals 1"
- For any nonzero base $a$: $$a^0=1 \quad (a\ne 0)$$
- Why this makes sense: using the quotient rule, $$\frac{a^m}{a^m}=a^{m-m}=a^0$$
- But $\frac{a^m}{a^m}=1$, so $a^0=1$.
- $0^0$ is not treated as a standard number in this context.
- For exponent laws at this level, remember $a^0=1$ requires $a\ne 0$.
Negative Exponents Mean "Reciprocal"
A negative exponent tells you to take the reciprocal: $$a^{-n}=\frac{1}{a^n} \quad (a\ne 0)$$
$$10^{-3}=\frac{1}{10^3}=\frac{1}{1000}=0.001$$
This connects directly to scientific notation, which we are going go talk about in the next article.
Three frequent errors:
- Adding exponents when bases are different: $2^3\cdot 5^3\ne 7^6$. Correct: $(2\cdot 5)^3=10^3$.
- Distributing exponents over addition: $(a+b)^2\ne a^2+b^2$.
- Forgetting restrictions: $\frac{a^m}{a^n}$ requires $a\ne 0$.
- Simplify: $\frac{a^7\,a^{-3}}{a^2}$.
- Evaluate: $(-3)^4$ and $-3^4$.
