ACT — mathMedium1 min read

Exponents and Square Roots

Apply exponent rules and simplify square roots for the ACT. Handle negative and fractional exponents.

Tutor AI Team
Tutor AI Team

Exponent and square root questions are common on the ACT. You need to know the exponent rules and how to simplify radicals.

Exponent Rules

Rule Formula Example
Product am⋅an=am+na^m \cdot a^n = a^{m+n} 23⋅24=272^3 \cdot 2^4 = 2^7
Quotient aman=am−n\frac{a^m}{a^n} = a^{m-n} 5652=54\frac{5^6}{5^2} = 5^4
Power (am)n=amn(a^m)^n = a^{mn} (32)3=36(3^2)^3 = 3^6
Zero a0=1a^0 = 1 70=17^0 = 1
Negative a−n=1ana^{-n} = \frac{1}{a^n} 2−3=182^{-3} = \frac{1}{8}
Fractional a1/n=ana^{1/n} = \sqrt[n]{a} 81/3=28^{1/3} = 2

Simplifying Square Roots

50=25×2=52\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}

72=36×2=62\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}

Operations with Radicals

32+52=823\sqrt{2} + 5\sqrt{2} = 8\sqrt{2} (combine like radicals).

3×12=36=6\sqrt{3} \times \sqrt{12} = \sqrt{36} = 6.

Practice Problems

    1. Simplify x5⋅x3x2\frac{x^5 \cdot x^3}{x^2}.
    1. Simplify 128\sqrt{128}.
    1. Evaluate 272/327^{2/3}.
    1. Simplify 12+27\sqrt{12} + \sqrt{27}.

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Key Takeaways

  • ✓

    Know all exponent rules — they're tested frequently.

  • ✓

    Simplify radicals by finding the largest perfect square factor.

  • ✓

    a−n=1ana^{-n} = \frac{1}{a^n} and a1/n=ana^{1/n} = \sqrt[n]{a}.

Tags:
#act#math#pre-algebra#exponents#square-roots

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