1. Laws of Exponents

You should know and be able to apply these rules quickly:

• Product Rule:
  $$a^m\cdota^n=a^{m+n}$$

• Quotient Rule:
  $$\frac{a^m}{a^n}=a^{m-n}$$

• Power of a Power Rule:
  $$(a^n)^m=a^{mn}$$

• Power of a Product Rule:
  $$(ab)^m=a^m\cdot b^m$$

• Power of a Quotient Rule:
  $$(\frac{a}{b})^m=\frac{a^m}{b^m}$$​

• Zero Exponent Rule:
  $$a^0=1,\ a\ne0$$

• Negative Exponent Rule:
  $$a^{-m}=\frac{1}{a^m}$$​

2. Evaluating Expressions with Exponents

• Be comfortable simplifying numbers like:

  •  $$2^3\cdot2^4=2^7$$

  •  $$\frac{3^5}{3^2}=3^3=27$$

• Know how to handle fractional exponents:

  • $$a^{1/2}=\sqrt{a}$$​

  • $$a^{m/n}=\sqrt[n]{a^m}$$​

• Example:
  $$64^{2/3}=(\sqrt[3]{64})^2=4^2=16$$

3. Solving Equations with Exponents

• If bases are the same, set exponents equal:

  • $$2^x=2^5 \to x=5$$

• If bases are different, rewrite if possible:

  • $$9^x=3^{2x}$$
    Since $$9=3^2$$, this becomes $$(3^2)^x=3^{2x}$$ (true for all real x).

• If not possible, use logarithms:

  • $$5^x=20 \to x=\log_{5}(20)$$

4. Simplifying Square Roots (Radicals)

• Break numbers into prime factors:

  • $$\sqrt{50}=\sqrt{25\cdot2}=5\sqrt{2}$$​

• Rationalize denominators:

  • $$\frac{1}{\sqrt3}=\frac{\sqrt{3}}{3}$$​​

• Combine radicals:

  • $$\sqrt{18}+\sqrt{8}=3\sqrt{2}+2\sqrt{2}=5\sqrt{2}$$​

5. Common ACT Tricks

• Watch out for hidden perfect squares:
  $$\sqrt{49x^2}=7|x|$$ (absolute value matters on the ACT).

• Fractional exponents vs radicals:
  $$\sqrt[4]{81}=81^{1/4}=3$$

• Always check for simplification opportunities in multiple-choice answers.

ACT Tip: Many ACT questions test your ability to apply exponent rules quickly or simplify radicals cleanly. Knowing these rules well not only saves time but also helps you avoid small mistakes. One effective way to build this skill is by practicing a range of problems, fundamental, intermediate, and advanced, like those included in the Exponents & Radicals section of the ACT Qbank from The School of Mathematics, which includes all exam-style questions from the past several years of the ACT test.