Exponents and Roots

What exponents mean

$b^n$ means $b$ multiplied by itself $n$ times.

$2^3 = 2 \times 2 \times 2 = 8$

• $b$ is the base; $n$ is the exponent or power.
• Special cases: $b^0 = 1$ (for any $b \neq 0$) and $b^1 = b$.

Laws of exponents

These five rules cover almost everything.

Rule	Example
$b^m \cdot b^n = b^{m+n}$	$2^3 \cdot 2^4 = 2^7$
$\dfrac{b^m}{b^n} = b^{m-n}$	$\dfrac{5^7}{5^4} = 5^3$
$(b^m)^n = b^{mn}$	$(3^2)^4 = 3^8$
$(ab)^n = a^n b^n$	$(2x)^3 = 8x^3$
$b^{-n} = \dfrac{1}{b^n}$	$2^{-3} = \dfrac{1}{8}$

Square roots

$\sqrt{x}$ asks: what positive number, squared, gives $x$? So $\sqrt{25} = 5$ because $5^2 = 25$.

A perfect square is the square of an integer: $1, 4, 9, 16, 25, 36, 49, 64, 81, 100, \ldots$

Simplifying radicals

If the number under the root has a perfect-square factor, you can pull it out.

Example. $\sqrt{72}$. Since $72 = 36 \times 2$:

$\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \cdot \sqrt{2} = 6\sqrt{2}$

Example. $\sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}$.

Fractional exponents

$b^{1/n}$ is the $n$-th root of $b$:

$b^{1/2} = \sqrt{b} \qquad b^{1/3} = \sqrt[3]{b}$

More generally, $b^{m/n} = \sqrt[n]{b^m}$.

Example. $8^{2/3} = \sqrt[3]{8^2} = \sqrt[3]{64} = 4$. (Or compute as $(\sqrt[3]{8})^2 = 2^2 = 4$.)

Key takeaways

• $b^n$ is $b$ multiplied by itself $n$ times; $b^0 = 1$.
• Multiplying same-base exponentials adds exponents; dividing subtracts.
• $\sqrt{xy} = \sqrt{x}\,\sqrt{y}$ — pull out perfect-square factors to simplify.
• Fractional exponent $m/n$ means $n$-th root of the $m$-th power.