Exponent Rule

Definition

An exponent rule is a mathematical principle that describes how to perform operations involving exponents. Exponents are used to represent repeated multiplication of a number, and exponent rules provide a systematic way to manipulate and simplify expressions containing exponents.

5 Must Know Facts For Your Next Test

1. Exponent rules allow for the simplification of complex expressions involving multiplication, division, and powers of the same base.
2. The product rule states that when multiplying powers with the same base, the exponents are added: $a^m \cdot a^n = a^{m+n}$.
3. The quotient rule states that when dividing powers with the same base, the exponents are subtracted: $\frac{a^m}{a^n} = a^{m-n}$.
4. The power rule states that when raising a power to a power, the exponents are multiplied: $(a^m)^n = a^{m\cdot n}$.
5. Rational exponents, such as $a^{\frac{1}{2}}$, represent the $\sqrt{a}$ or square root of $a$, and follow the same exponent rules.

Review Questions

• Explain how the product rule for exponents can be used to simplify an expression.
  • The product rule for exponents states that when multiplying powers with the same base, the exponents are added. For example, if we have the expression $2^3 \cdot 2^4$, we can use the product rule to simplify it to $2^{3+4} = 2^7$. This rule allows us to condense repeated multiplication of the same base into a single term with the sum of the exponents.
• Describe how the quotient rule for exponents can be used to simplify an expression involving division.
  • The quotient rule for exponents states that when dividing powers with the same base, the exponents are subtracted. For instance, if we have the expression $\frac{5^8}{5^3}$, we can use the quotient rule to simplify it to $5^{8-3} = 5^5$. This rule enables us to cancel out common factors in the numerator and denominator of a fraction containing exponents.
• Analyze how the power rule for exponents can be applied to expressions involving raising a power to a power.
  • The power rule for exponents states that when raising a power to a power, the exponents are multiplied. For example, if we have the expression $(2^3)^4$, we can use the power rule to simplify it to $2^{3\cdot 4} = 2^{12}$. This rule allows us to condense nested exponents into a single term by multiplying the exponents together. Understanding the power rule is crucial for working with rational exponents and simplifying complex expressions involving exponents.

Related terms

Power: The exponent, or the number that represents how many times the base is multiplied by itself.

Base: The number that is being raised to a power.

Radical: The symbol used to represent a root, which is the inverse operation of raising a number to a power.