key term - Laws of Exponents

Definition

The laws of exponents are a set of rules that govern the manipulation and simplification of expressions involving exponents. These laws are fundamental to understanding and working with radicals and rational exponents, which are crucial topics in college algebra.

5 Must Know Facts For Your Next Test

1. The product rule states that $a^m \cdot a^n = a^{m+n}$, where $a$ is the base and $m$ and $n$ are the exponents.
2. The power rule states that $(a^m)^n = a^{m \cdot n}$, where $a$ is the base and $m$ and $n$ are the exponents.
3. The quotient rule states that $\frac{a^m}{a^n} = a^{m-n}$, where $a$ is the base and $m$ and $n$ are the exponents.
4. The zero exponent rule states that $a^0 = 1$, where $a$ is any non-zero base.
5. The negative exponent rule states that $a^{-n} = \frac{1}{a^n}$, where $a$ is the base and $n$ is the exponent.

Review Questions

• Explain how the product rule of exponents can be used to simplify expressions involving multiplication of terms with the same base.
  • The product rule of exponents states that when you multiply terms with the same base, you can add the exponents together. For example, $a^m \cdot a^n = a^{m+n}$. This rule allows you to simplify expressions like $2^3 \cdot 2^4 = 2^{3+4} = 2^7$. By applying the product rule, you can combine the exponents and reduce the expression to a single term with a simpler exponent.
• Describe how the power rule of exponents can be used to evaluate expressions with exponents raised to other exponents.
  • The power rule of exponents states that when you have an expression with an exponent raised to another exponent, you can multiply the exponents together. For example, $(a^m)^n = a^{m \cdot n}$. This rule allows you to simplify expressions like $(2^3)^4 = 2^{3 \cdot 4} = 2^{12}$. By applying the power rule, you can evaluate the inner exponent first and then multiply the exponents together to arrive at the final simplified expression.
• Analyze how the laws of exponents, specifically the quotient rule and the negative exponent rule, can be used to simplify expressions involving division and negative exponents.
  • The quotient rule of exponents states that when you divide terms with the same base, you can subtract the exponents. For example, $\frac{a^m}{a^n} = a^{m-n}$. Additionally, the negative exponent rule states that $a^{-n} = \frac{1}{a^n}$. These rules allow you to simplify expressions involving division and negative exponents. For instance, $\frac{2^5}{2^3} = 2^{5-3} = 2^2 = 4$, and $(2^3)^{-2} = 2^{3 \cdot (-2)} = 2^{-6} = \frac{1}{2^6} = \frac{1}{64}$. By applying these laws, you can transform complex expressions with division and negative exponents into simpler, more manageable forms.

Related terms

Exponent:

A number that represents the power to which a base number is raised, indicating how many times the base is multiplied by itself.

Radical:

The symbol (√) used to represent a root of a number, often a square root or cube root.

Rational Exponent:

An exponent that can be expressed as a fraction, representing a root of a number.