Product Property of Exponents

Definition

The product property of exponents states that when multiplying two expressions with the same base, the exponents can be added together. This property simplifies the multiplication of expressions with the same base.

5 Must Know Facts For Your Next Test

1. The product property of exponents allows you to multiply expressions with the same base by adding the exponents.
2. This property is useful when dividing monomials, as it can simplify the division process.
3. The product property of exponents can be expressed mathematically as: $a^m \cdot a^n = a^{m+n}$.
4. This property holds true for both positive and negative exponents.
5. Understanding the product property of exponents is crucial for simplifying and manipulating exponential expressions in algebra.

Review Questions

• Explain how the product property of exponents can be used to simplify the division of monomials.
  • The product property of exponents states that when multiplying expressions with the same base, the exponents can be added together. This property can be applied to the division of monomials by rewriting the division as a multiplication with a negative exponent. For example, to divide $a^5$ by $a^3$, you can rewrite it as $a^5 \div a^3 = a^{5-3} = a^2$. This simplifies the division process by subtracting the exponents instead of performing long division.
• Describe the mathematical expression that represents the product property of exponents and explain its significance.
  • The product property of exponents can be expressed mathematically as: $a^m \cdot a^n = a^{m+n}$. This expression shows that when multiplying two expressions with the same base, the exponents can be added together to simplify the expression. This property is significant because it allows for the efficient manipulation and simplification of exponential expressions, which is crucial in various algebraic operations, such as division, factorization, and simplification of complex expressions.
• Analyze how the product property of exponents can be applied to expressions with both positive and negative exponents, and explain the implications of this property in the context of dividing monomials.
  • The product property of exponents holds true for both positive and negative exponents. This means that the property can be applied to expressions with any combination of positive and negative exponents. For example, $a^{-2} \cdot a^3 = a^{-2+3} = a^1 = a$. The ability to apply the product property to expressions with negative exponents is particularly useful in the context of dividing monomials. When dividing monomials, you can rewrite the division as a multiplication with a negative exponent, which simplifies the process by subtracting the exponents instead of performing long division. This property is essential for efficiently manipulating and simplifying exponential expressions in the context of dividing monomials.

Related terms

Exponent:

A mathematical notation that indicates the number of times a value (the base) is multiplied by itself.

Base:

The number or variable that is being raised to a power in an exponential expression.

Power:

The exponent that indicates how many times the base is multiplied by itself.