Exponent Properties

Definition

Exponent properties are the rules and relationships that govern the behavior of exponents, which are the small numbers written above and to the right of a base number. These properties allow for the simplification and manipulation of expressions involving exponents, and are particularly important in the context of rational exponents.

5 Must Know Facts For Your Next Test

1. The product of powers with the same base is the base raised to the sum of the exponents: $a^m \cdot a^n = a^{m+n}$.
2. The quotient of powers with the same base is the base raised to the difference of the exponents: $\frac{a^m}{a^n} = a^{m-n}$.
3. The power of a power is the base raised to the product of the exponents: $(a^m)^n = a^{m\cdot n}$.
4. The power of a product is the product of the powers: $(a \cdot b)^n = a^n \cdot b^n$.
5. The power of a quotient is the quotient of the powers: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$.

Review Questions

• Explain how the product of powers with the same base can be simplified using exponent properties.
  • The product of powers with the same base can be simplified by adding the exponents. For example, $a^m \cdot a^n = a^{m+n}$. This property allows for the efficient simplification of expressions involving the multiplication of powers with the same base, which is particularly useful when working with rational exponents.
• Describe the relationship between the power of a power and the product of the exponents.
  • The power of a power property states that $(a^m)^n = a^{m\cdot n}$. This means that when raising a power to another power, the exponents are multiplied. This property can be used to simplify complex expressions involving nested exponents, which is important when working with rational exponents and understanding the behavior of exponents in general.
• Analyze how the power of a quotient property can be used to rewrite expressions involving fractions with exponents.
  • The power of a quotient property states that $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$. This allows you to rewrite expressions involving fractions with exponents by moving the exponent to the numerator and denominator separately. This is a crucial property for simplifying and manipulating rational exponent expressions, as it enables you to work with the numerator and denominator independently.

Related terms

Base: The number or variable that is raised to an exponent, also known as the base.

Exponent: The small number written above and to the right of a base, indicating the number of times the base is multiplied by itself.

Power: The result of raising a base to an exponent, also known as the power.