Foundations Reference
Exponents and Radicals
Exponent rules keep repeated multiplication organized. Negative exponents move factors across a fraction bar, and fractional exponents connect powers with roots.
Exponent Rules Table
| Rule | Formula | Meaning |
|---|---|---|
| Product rule | $a^m a^n=a^{m+n}$ | Same base multiplied: add exponents. |
| Quotient rule | $\frac{a^m}{a^n}=a^{m-n}$ | Same base divided: subtract exponents. |
| Power rule | $(a^m)^n=a^{mn}$ | Power raised to a power: multiply exponents. |
| Power of a product | $(ab)^n=a^n b^n$ | The outside exponent applies to each factor. |
| Zero exponent | $a^0=1$ | Any nonzero base to the zero power is $1$. |
| Negative exponent | $a^{-n}=\frac1{a^n}$ | A negative exponent means reciprocal, not negative value. |
Exponent Formulas
Negative Exponent
$$a^{-n}=\frac1{a^n}$$ $$\frac1{a^{-n}}=a^n$$
Fractional Exponent
$$a^{1/n}=\sqrt[n]{a}$$
Power Over Root
$$a^{m/n}=\sqrt[n]{a^m}=\left(\sqrt[n]{a}\right)^m$$
Even Root Domain
$$\sqrt[n]{a}\text{ is real only when }a\ge0\text{ if }n\text{ is even}$$
The denominator of a fractional exponent is the root. The numerator is the power.
Classic Examples
Product and Quotient Rules
Simplify $\frac{x^7x^3}{x^4}$.
Exponent MoveAdd exponents when multiplying the same base, then subtract when dividing.
$$\frac{x^7x^3}{x^4}$$
$$=\frac{x^{10}}{x^4}$$
$$=x^6$$
Power of a Product
Simplify $(3x^2y^{-1})^3$ using positive exponents.
Exponent MoveThe outside exponent applies to every factor inside the parentheses.
$$(3x^2y^{-1})^3$$
$$=3^3x^{6}y^{-3}$$
$$=\frac{27x^6}{y^3}$$
Negative Exponents
Simplify $\frac{4a^{-2}b^3}{8ab^{-1}}$ using positive exponents.
Exponent MoveSubtract exponents by base, then move factors with negative exponents across the fraction bar.
$$\frac{4a^{-2}b^3}{8ab^{-1}}$$
$$=\frac12a^{-3}b^4$$
$$=\frac{b^4}{2a^3}$$
Fractional Exponent
Evaluate $27^{2/3}$.
Exponent MoveThe denominator $3$ tells us to take a cube root; the numerator $2$ tells us to square.
$$27^{2/3}$$
$$=(\sqrt[3]{27})^2$$
$$=3^2$$
$$=9$$
Radical to Exponent Form
Rewrite $\sqrt[5]{x^3}$ using rational exponents.
Exponent MoveThe root index becomes the denominator of the exponent.
$$\sqrt[5]{x^3}$$
$$=x^{3/5}$$
Negative Fractional Exponent
Simplify $16^{-3/4}$.
Exponent MoveThe negative exponent makes a reciprocal; the denominator $4$ asks for a fourth root.
$$16^{-3/4}$$
$$=\frac{1}{16^{3/4}}$$
$$=\frac{1}{(\sqrt[4]{16})^3}$$
$$=\frac{1}{2^3}$$
$$=\frac18$$