M151Q Precalculus · Unit 2: Powers, exponentials, logarithms · on Exam 2
MFG 3.2–3.4 Exponents, roots, radicals, and rational exponents
Study notes for Exponents, roots, radicals, and rational exponents in M151Q Precalculus at Montana State University: the big ideas, key formulas, a worked example, common mistakes and an exam tip. The same topic has endless practice questions and flashcards in Mathub.
Big ideas
- Laws: $a^ma^n = a^{m+n}$, $a^m/a^n = a^{m-n}$, $(a^m)^n = a^{mn}$, $(ab)^n = a^nb^n$, $a^0 = 1$, $a^{-n} = 1/a^n$. Negative exponents mean reciprocals, not negative numbers.
- Radicals are exponents: $\sqrt[n]{a} = a^{1/n}$ and $a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$. Evaluate $8^{2/3}$ as $(\sqrt[3]{8})^2 = 4$.
- Even roots of negatives are not real; odd roots are fine. $\sqrt{x^2} = |x|$.
- Solve power equations by isolating the power and applying the reciprocal exponent: $x^{3/2} = 27 \Rightarrow x = 27^{2/3} = 9$. Check for extraneous solutions with even powers.
- Simplify radicals by pulling out perfect powers; rationalize denominators when asked.
Key formulas
Laws of exponents
$$a^ma^n = a^{m+n},\quad \frac{a^m}{a^n} = a^{m-n},\quad (a^m)^n = a^{mn},\quad a^{-n} = \frac1{a^n},\quad a^0 = 1$$
Rational exponents
$$a^{1/n} = \sqrt[n]{a},\qquad a^{m/n} = \left(\sqrt[n]{a}\right)^m$$
Power equations
$$x^{p} = k \Rightarrow x = k^{1/p}\ (\text{with } \pm \text{ if } p \text{ has even numerator})$$
Worked example
Simplify $\dfrac{(2x^3)^2\,x^{-4}}{4x}$ and evaluate $16^{-3/4}$.
Show the solution
$\dfrac{4x^6x^{-4}}{4x} = \dfrac{4x^2}{4x} = x$. $16^{-3/4} = \dfrac{1}{(\sqrt[4]{16})^3} = \dfrac{1}{2^3} = \dfrac18$.
Common mistakes
- $a^{-2} = -a^2$ (wrong). It is $1/a^2$.
- $(a+b)^2 = a^2 + b^2$ (wrong).
- Taking $\sqrt{x^2 + 9}$ as $x + 3$.
Exam tipRewrite every radical as a rational exponent before simplifying; the exponent laws then do all the work.
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