$(a^m)^n$

Definition

The expression $(a^m)^n$ represents the result of raising the power $a^m$ to the power of $n$. This is an example of the exponent rules, specifically the power rule, which states that $(a^m)^n = a^{m \times n}$. This allows for simplification and manipulation of exponents in algebraic expressions.

5 Must Know Facts For Your Next Test

1. The power rule, $(a^m)^n = a^{m \times n}$, allows us to simplify expressions involving exponents raised to other exponents.
2. When applying the power rule, the exponents are multiplied: the exponent of the base ($a$) is multiplied by the exponent of the power ($n$).
3. Simplifying expressions with the power rule can help reduce the complexity of algebraic expressions and make them easier to work with.
4. The power rule is an important concept in the context of 6.2 Use Multiplication Properties of Exponents, as it allows for the manipulation and simplification of products and quotients of powers.
5. Understanding the power rule and being able to apply it correctly is crucial for solving a variety of algebraic problems involving exponents.

Review Questions

• Explain the power rule, $(a^m)^n = a^{m \times n}$, and how it can be used to simplify expressions with exponents.
  • The power rule states that when a power is raised to another power, the exponents are multiplied. Specifically, $(a^m)^n = a^{m \times n}$. This rule allows us to simplify expressions with exponents raised to other exponents by multiplying the exponents together. For example, $(a^3)^4 = a^{3 \times 4} = a^{12}$. Applying the power rule helps reduce the complexity of algebraic expressions and makes them easier to work with, which is an important skill in the context of 6.2 Use Multiplication Properties of Exponents.
• Describe how the power rule, $(a^m)^n = a^{m \times n}$, relates to the multiplication properties of exponents covered in 6.2.
  • The power rule, $(a^m)^n = a^{m \times n}$, is closely connected to the multiplication properties of exponents covered in 6.2. Specifically, the power rule allows us to simplify expressions involving exponents raised to other exponents, which is a key skill for understanding and applying the multiplication properties of exponents. By being able to apply the power rule, we can more easily manipulate and simplify products and quotients of powers, which is a crucial concept in 6.2 Use Multiplication Properties of Exponents.
• Analyze how the power rule, $(a^m)^n = a^{m \times n}$, can be used to solve a variety of algebraic problems involving exponents.
  • The power rule, $(a^m)^n = a^{m \times n}$, is a versatile tool that can be applied to solve a wide range of algebraic problems involving exponents. By understanding and correctly applying this rule, students can simplify complex expressions, evaluate numerical expressions with exponents, and manipulate exponents in equations and inequalities. The power rule is foundational for working with exponents, and mastering its application is essential for success in 6.2 Use Multiplication Properties of Exponents and beyond. Skillful use of the power rule demonstrates a deep understanding of exponent rules and the ability to apply them to solve diverse algebraic problems.

Related terms

Exponent: An exponent is a number that indicates how many times the base number is multiplied by itself. For example, in $a^m$, the exponent $m$ tells us how many times $a$ is multiplied by itself.

Power Rule: The power rule is an exponent rule that states $(a^m)^n = a^{m \times n}$. This allows us to simplify expressions involving exponents raised to other exponents.

Simplifying Exponents: Simplifying exponents involves applying the various exponent rules, such as the power rule, to reduce the expression to its simplest form.