Basics of Exponents

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An exponent is the power a number or variable is raised to in an expression. This power can be positive, negative, zero, and fractional. Indeed, the exponent can be any real number. If the exponent is 0, then the value reduces to 1; that is, \(\begin{align}x^0 = 1\end{align}\), for every non-zero real number x. Other examples:

• \(\begin{align}2^3 = 2 \times 2 \times 2 = 8\end{align}\)
• \(\begin{align}2^{-3} = \frac{1}{2^3} = \frac{1}{8}\end{align}\)
• \(\begin{align}2^\frac{1}{2} = \sqrt{2}\end{align}\)

Rules of Exponents

Exponent problems can usually be solved by manipulating the component parts according to the rules of exponents, which we summarize as:

1. x^a ∙ x^b = x^a+b Example: 2³ ∙ 2⁴ = 2⁷, or 8 ∙ 16 = 128 = 2⁷
2. (x^a)^b = x^ab Example: (3²)³ = 3^2∙3 = 3⁶, or 9³ = 729, and 3⁶ = 729
3. x^a / x^b = x^a-b Example: 2⁴/2³ = 2^4-3 = 2¹, or 16/8 = 2
4. x^-a = 1/x^a Example: 2^-3 = 1/2³ = 1/8
5. 1/x^-a = x^a Example: 1/3^-2 = 3² = 9
6. \(\begin{align}x^\frac{m}{n} = \sqrt[n]{x^m}\end{align}\) Example: \(\begin{align}2^\frac{2}{3} = \sqrt[3]{2^2} = \sqrt[3]{4}\end{align}\)
7. x^a ∙ y^a = (xy)^a Example: 3² · 4² = (3 · 4)² = 12² = 144 = 9 · 16 = 3² · 4²

Sample Exponent Problems