Law of Exponents

Definition

The law of exponents is a fundamental principle in algebra that governs the rules for manipulating and simplifying expressions involving exponents. It provides a consistent set of guidelines for performing operations such as multiplication, division, and raising to a power with exponents.

5 Must Know Facts For Your Next Test

1. The law of exponents states that when dividing monomials with the same base, the exponents are subtracted.
2. When multiplying monomials with the same base, the exponents are added.
3. When raising a power to a power, the exponents are multiplied.
4. The law of exponents applies to both positive and negative exponents.
5. Understanding the law of exponents is crucial for simplifying and manipulating algebraic expressions, especially in the context of dividing monomials.

Review Questions

• Explain how the law of exponents is used when dividing monomials.
  • When dividing monomials with the same base, the law of exponents states that the exponents are subtracted. For example, to divide $x^5$ by $x^3$, the exponents are subtracted: $x^5 \div x^3 = x^{5-3} = x^2$. This rule allows for the simplification of complex monomial expressions involving division.
• Describe the relationship between the law of exponents and the multiplication of monomials.
  • According to the law of exponents, when multiplying monomials with the same base, the exponents are added. For instance, to multiply $x^4$ and $x^3$, the exponents are added: $x^4 \times x^3 = x^{4+3} = x^7$. This property is crucial for efficiently simplifying and manipulating algebraic expressions involving the multiplication of monomials.
• Analyze how the law of exponents can be applied when raising a power to a power.
  • The law of exponents states that when raising a power to a power, the exponents are multiplied. For example, to evaluate $(x^5)^3$, the exponents are multiplied: $(x^5)^3 = x^{5 \times 3} = x^{15}$. This rule allows for the simplification of complex expressions involving the raising of powers, which is particularly useful when working with monomials and their operations.

Related terms

Exponent: A number that indicates how many times a base number is multiplied by itself. Exponents are represented using a small number written to the right of and above the base number.

Power: The result of raising a number or expression to an exponent. For example, $5^3$ is read as 'five to the power of three' and represents the product of multiplying 5 by itself 3 times.

Monomial: An algebraic expression with a single term, consisting of a coefficient, variables, and exponents.