Exponent Calculator

What is an Exponent?

An exponent is a mathematical notation representing repeated multiplication of a base number. For example, $a^n$ means the number $a$ (the base) is multiplied by itself $n$ times. Exponents simplify complex calculations, especially in areas like algebra, geometry, and calculus. For instance:

$$3^4 = 3 \times 3 \times 3 \times 3 = 81$$

Exponents can also extend to fractional and negative numbers, allowing advanced mathematical operations like finding roots or reciprocals.

Main Exponent Rules and Properties

1. Product of Powers Rule: When multiplying with the same base, add the exponents.

   $$a^m \cdot a^n = a^{m+n}$$

   Example: $2^3 \cdot 2^2 = 2^{3+2} = 2^5 = 32$.

2. Quotient of Powers Rule: When dividing with the same base, subtract the exponents.

   $$\frac{a^m}{a^n} = a^{m-n}, \, \text{where } m > n$$

   Example: $\frac{5^3}{5^1} = 5^{3-1} = 5^2 = 25$.

3. Power of a Power Rule: When raising an exponent to another exponent, multiply them.

   $$(a^m)^n = a^{m \cdot n}$$

   Example: $(2^3)^2 = 2^{3 \cdot 2} = 2^6 = 64$.

4. Zero Exponent Rule: Any non-zero base raised to the power of 0 equals 1.

   $$a^0 = 1$$

   Example: $7^0 = 1$.

5. Negative Exponent Rule: A negative exponent indicates reciprocal.

   $$a^{-n} = \frac{1}{a^n}$$

   Example: $2^{-2} = \frac{1}{2^2} = \frac{1}{4}$.

6. Fractional Exponent Rule: A fractional exponent represents a root.

   $$a^{\frac{m}{n}} = \sqrt[n]{a^m}$$

   Example: $8^{\frac{1}{3}} = \sqrt[3]{8} = 2$.

7. Distributive Properties:

   • For multiplication: $$(ab)^n = a^n \cdot b^n$$
   • For division: $$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$$

Sources:

Larson, R., Falvo, D.C. and Hostetler, R.P. (2010). Precalculus with Limits., pp. 216-228.