Powers and Polynomials 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.

Solved Examples

Jump directly to a worked example by exponent or radical type.

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 $\frac{x^7x^3}{x^4}$.

Solution Steps
  1. Combine the numerator powers by adding exponents.
  2. Divide powers with the same base by subtracting exponents.
  3. State the simplified expression: $x^6$.
$$\frac{x^7x^3}{x^4}$$ $$=\frac{x^{10}}{x^4}$$ $$=x^6$$

Power of a Product

Simplify $(3x^2y^{-1})^3$ using positive exponents.

Solution Steps
  1. Apply the outside exponent to every factor.
  2. Multiply the exponents on each variable.
  3. Move the negative power to the denominator and state the result.
$$(3x^2y^{-1})^3$$ $$=3^3x^{6}y^{-3}$$ $$=\frac{27x^6}{y^3}$$

Negative Exponents

Simplify $\frac{4a^{-2}b^3}{8ab^{-1}}$ using positive exponents.

Solution Steps
  1. Separate the numerical factor and subtract exponents by base.
  2. Write the resulting negative powers.
  3. Move negative-power factors across the fraction bar.
$$\frac{4a^{-2}b^3}{8ab^{-1}}$$ $$=\frac12a^{-3}b^4$$ $$=\frac{b^4}{2a^3}$$

Fractional Exponent

Evaluate $27^{2/3}$.

Solution Steps
  1. Read the denominator as a cube root.
  2. Apply the numerator as a square.
  3. Evaluate the cube root.
  4. State the value: $9$.
$$27^{2/3}$$ $$=(\sqrt[3]{27})^2$$ $$=3^2$$ $$=9$$

Radical to Exponent Form

Rewrite $\sqrt[5]{x^3}$ using rational exponents.

Solution Steps
  1. Use the root index as the exponent denominator.
  2. State the rational-exponent form: $x^{3/5}$.
$$\sqrt[5]{x^3}$$ $$=x^{3/5}$$

Negative Fractional Exponent

Simplify $16^{-3/4}$.

Solution Steps
  1. Rewrite the negative exponent as a reciprocal.
  2. Read the denominator as a fourth root.
  3. Evaluate the fourth root.
  4. Evaluate the power of $2$.
  5. State the result: $\frac18$.
$$16^{-3/4}$$ $$=\frac{1}{16^{3/4}}$$ $$=\frac{1}{(\sqrt[4]{16})^3}$$ $$=\frac{1}{2^3}$$ $$=\frac18$$