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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.
A common mistake is confusing $-2^4$ with $(-2)^4$.
Use parentheses whenever the base is negative.
$$4^3\times 4^8=4^{3+8}=4^{11}$$
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: $3^5\times 2^5=(3\cdot 2)^5=6^5$.
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 $3^5\times(-2)^5$: $$3^5\times(-2)^5=(3\cdot -2)^5=(-6)^5$$
When dividing powers with the same base: $$\frac{a^m}{a^n}=a^{m-n} \quad (a\ne 0)$$
$$\frac{12^{15}}{12^2}=12^{15-2}=12^{13}$$
$$\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n} \quad (b\ne 0)$$
This matches the textbook form: $$\frac{a^n}{b^n}=\left(\frac{a}{b}\right)^n$$
$$\frac{(-48)^7}{4^7}=\left(\frac{-48}{4}\right)^7=(-12)^7$$
$$\left(a^m\right)^n=a^{mn}$$
$$\left(2^4\right)^3=2^{4\cdot 3}=2^{12}$$
Exponent laws are most powerful when we extend them beyond positive integers.
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: