Properties of Integer Exponents

Exponents are a shorthand way to show repeated multiplication. When working with algebra, understanding the properties of integer exponents allows you to easily simplify complex expressions. These rules apply to all integer powers, whether they are positive, negative, or zero.

Fundamental Rules of Exponents

Here are the core properties you need to know when working with the same base:

• Product Rule: When multiplying terms with the same base, add the exponents. $a^m \cdot a^n = a^{m+n}$
• Quotient Rule: When dividing terms with the same base, subtract the exponents. $\frac{a^m}{a^n} = a^{m-n}$
• Power of a Power Rule: When raising a power to another power, multiply the exponents. $(a^m)^n = a^{m \cdot n}$
• Power of a Product Rule: A power applied to a product is distributed to each factor. $(ab)^m = a^m b^m$
• Power of a Quotient Rule: A power applied to a fraction is distributed to both the numerator and denominator. $\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}$

Zero and Negative Exponents

Exponents aren't always positive integers. Here is how to handle zero and negative powers:

• Zero Exponent Rule: Any non-zero number raised to the power of zero is exactly $1$. $a^0 = 1 \quad (a \neq 0)$
• Negative Exponent Rule: A negative exponent means to take the reciprocal of the base and make the exponent positive. $a^{-n} = \frac{1}{a^n} \quad (a \neq 0)$

Example Problems

Example 1: Simplify $(3x^2 y^3)^2 \cdot (2xy)^{-1}$

1. First, apply the Power of a Product Rule to the first term: $(3x^2 y^3)^2 = 3^2 \cdot (x^2)^2 \cdot (y^3)^2 = 9x^4 y^6$
2. Next, apply the Negative Exponent Rule to the second term: $(2xy)^{-1} = \frac{1}{2xy}$
3. Multiply them together and apply the Quotient Rule: $\frac{9x^4 y^6}{2xy} = \frac{9}{2} x^{4-1} y^{6-1} = \frac{9}{2} x^3 y^5$

Example 2: Write $0.00045$ in scientific notation

Scientific notation uses integer exponents to represent very large or very small numbers. To write $0.00045$, move the decimal point $4$ places to the right to get a number between $1$ and $10$. Because we moved the decimal to the right (representing a small number), the exponent is negative: $0.00045 = 4.5 \times 10^{-4}$