Exponents and Radicals

Use exponent laws precisely

For the same base $a, a^m$ * $a^n = a^(m+n)$ and $a^m / a^n = a^(m-n)$ when a is nonzero. A power raised to a power multiplies exponents: $(a^m)^n = a^($mn).

Do not distribute exponents over addition$. (a + b)^2$ is not $a^2 + b^2$.

Connect radicals and rational exponents

The nth root of a can be written as $a^(1/n)$. So $\sqrt{a}$ is $a^(1/2)$, and cube root of a is $a^(1/3)$. This notation helps when multiplying and dividing radical expressions.

• $\sqrt{50}$ = $\sqrt{25*2}$ = 5sqrt(2).
• $a^(-3) = 1/a^3$ for a not equal to zero.
• $a^(m/n)$ means nth root of $a^m$.

Simplify an expression with exponents and radicals

Simplify $(18x^5y^2)/(3x^2y)$ and then rewrite $\sqrt{x^6}$ for x >= 0.

1. Divide coefficients: 18/3 = 6.
2. Subtract exponents on like bases: $x^(5-2) = x^3$ and $y^(2-1) = y$.
3. Simplified fraction is $6x^3y$.
4. For x >= 0, $\sqrt{x^6}$ $= x^3$ because $(x^3)^2 = x^6$.

Common mistakes

• Adding exponents across unlike bases.
• Treating a negative exponent as making the value negative.
• Assuming $\sqrt{a+b}$ equals $\sqrt{a}$ + $\sqrt{b}$.
• Dropping domain conditions when simplifying even roots.

Try one

Sources

• OpenStax: College AlgebraOpen textbook with full chapter coverage, worked examples, and practice problems.
• Khan Academy: Algebra and Algebra 2Official lessons and practice sets for core algebra skills.