Laws of Exponents

Definition

The laws of exponents are a set of rules that govern the operations and manipulations of exponents in mathematical expressions. These laws provide a consistent and logical framework for simplifying and evaluating expressions involving powers or exponents.

5 Must Know Facts For Your Next Test

1. The laws of exponents allow for the simplification and manipulation of expressions involving multiplication, division, and raising exponents to exponents.
2. The product rule states that when multiplying numbers with the same base, the exponents are added: $a^m \cdot a^n = a^{m+n}$.
3. The power rule states that when raising a power to a power, the exponents are multiplied: $(a^m)^n = a^{m \cdot n}$.
4. The quotient rule states that when dividing numbers with the same base, the exponents are subtracted: $\frac{a^m}{a^n} = a^{m-n}$.
5. The zero exponent rule states that any number raised to the power of 0 is equal to 1, except for 0 raised to the power of 0, which is undefined.

Review Questions

• Explain the product rule for exponents and provide an example of its application.
  • The product rule for exponents states that when multiplying numbers with the same base, the exponents are added. For example, if we have $a^3 \cdot a^5$, we can apply the product rule to simplify this expression as $a^{3+5} = a^8$. This rule allows us to combine repeated factors with the same base by adding the exponents, making the expression more concise and easier to evaluate.
• Describe the power rule for exponents and demonstrate how it can be used to simplify an expression.
  • The power rule for exponents states that when raising a power to a power, the exponents are multiplied. For instance, if we have $(a^4)^3$, we can apply the power rule to simplify this expression as $a^{4 \cdot 3} = a^{12}$. This rule is useful when dealing with nested exponents, as it allows us to combine the exponents by multiplying them, resulting in a more compact and manageable expression.
• Analyze the quotient rule for exponents and explain how it can be used to simplify an expression involving division of powers.
  • The quotient rule for exponents states that when dividing numbers with the same base, the exponents are subtracted. For example, if we have $\frac{a^7}{a^3}$, we can apply the quotient rule to simplify this expression as $a^{7-3} = a^4$. This rule is particularly useful when dealing with division of powers, as it allows us to cancel out common factors by subtracting the exponents, leading to a more streamlined and simplified expression.

Related terms

Exponent: A mathematical notation that represents repeated multiplication of a base number by itself. The exponent indicates the number of times the base is used as a factor.

Power: The result of raising a number or expression to an exponent, representing repeated multiplication of the base.

Base: The number or expression that is raised to an exponent, representing the factor that is multiplied repeatedly.