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Unit 5: Exponents, Roots, and Rational Expressions

Topic 2

Negative and Fractional Exponents

Exponents can be extended beyond positive integers. A negative exponent represents the reciprocal of a base raised to a positive power:
a−n=1an,a≠0.a^{-n} = \frac{1}{a^n}, \quad a \ne 0.
A fractional exponent represents a root:
a1n=an,amn=amn=(an)m.a^{\frac{1}{n}} = \sqrt[n]{a}, \quad a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m.
These definitions make the laws of exponents consistent for all rational exponents.
Understanding why these work helps connect exponent and root operations. For example, a12a^{\frac{1}{2}} must mean “the number that, when squared, gives aa” so that a12⋅2=a1a^{\frac{1}{2}\cdot 2} = a^1. Similarly, a−na^{-n} keeps the multiplication rule valid because am⋅a−m=am−m=a0=1.a^m \cdot a^{-m} = a^{m-m} = a^0 = 1.

Core Skills

  • Rewrite negative exponents as reciprocals.
  • Rewrite fractional exponents as roots.
  • Simplify expressions combining integer, negative, and fractional exponents.
  • Convert between radical and exponential forms.
  • Apply exponent laws consistently across all exponent types.

Example 1: Negative Exponent as Reciprocal

Simplify x−3x^{-3}.
x−3=1x3x^{-3} = \frac{1}{x^3}
Interpretation: A negative exponent inverts the base. Final Answer: 1x3\boxed{\dfrac{1}{x^3}}

Example 2: Fractional Exponent as Root

Simplify 161216^{\frac{1}{2}}.
1612=16=416^{\frac{1}{2}} = \sqrt{16} = 4
Interpretation: The denominator 2 indicates a square root. Final Answer: 4\boxed{4}

Example 3: Mixed Fractional Exponent

Simplify 272327^{\frac{2}{3}}.
2723=(273)2=32=927^{\frac{2}{3}} = (\sqrt[3]{27})^2 = 3^2 = 9
Interpretation: Cube root first, then square. Final Answer: 9\boxed{9}

Example 4: Negative and Fractional Combined

Simplify 8−238^{-\frac{2}{3}}.
8−23=1823=1(83)2=148^{-\frac{2}{3}} = \frac{1}{8^{\frac{2}{3}}} = \frac{1}{(\sqrt[3]{8})^2} = \frac{1}{4}
Final Answer: 14\boxed{\dfrac{1}{4}}

Example 5: Algebraic Expression with Mixed Exponents

Simplify x32x12\dfrac{x^{\frac{3}{2}}}{x^{\frac{1}{2}}}.
x32−12=x1=xx^{\frac{3}{2} - \frac{1}{2}} = x^{1} = x
Final Answer: x\boxed{x}

Example 6: Converting Between Radical and Exponential Form

Rewrite x34\sqrt[4]{x^3} using exponents.
x34=x34\sqrt[4]{x^3} = x^{\frac{3}{4}}
Final Answer: x34\boxed{x^{\frac{3}{4}}}

Key Takeaways

  • a−n=1ana^{-n} = \frac{1}{a^n} and a1n=ana^{\frac{1}{n}} = \sqrt[n]{a}.
  • Exponent rules (am⋅an=am+na^m \cdot a^n = a^{m+n}, etc.) hold for all rational exponents.
  • Negative means “reciprocal”; fractional means “root.”
  • Simplify by rewriting all terms with positive exponents before combining.
  • When in doubt, rewrite radicals as fractional powers for consistency.