key term - $a^n imes b^n$

Definition

$a^n imes b^n$ is a mathematical expression that represents the multiplication of two powers with the same exponent. It is a fundamental concept in the context of using the multiplication properties of exponents, which allow for the simplification and manipulation of expressions involving exponents.

5 Must Know Facts For Your Next Test

1. The expression $a^n imes b^n$ can be simplified using the multiplication property of exponents to $a^n imes b^n = (a imes b)^n$.
2. The exponent $n$ in the expression $a^n imes b^n$ must be the same for both $a$ and $b$ in order to apply the multiplication property of exponents.
3. The multiplication property of exponents allows for the simplification of expressions involving the multiplication of powers with the same exponent, making calculations more efficient.
4. Understanding the concept of $a^n imes b^n$ is crucial for manipulating and solving problems involving exponents, which are commonly encountered in algebra and other mathematical contexts.
5. The ability to apply the multiplication property of exponents to the expression $a^n imes b^n$ demonstrates a deeper understanding of the rules and properties governing exponents.

Review Questions

• Explain how the expression $a^n imes b^n$ can be simplified using the multiplication property of exponents.
  • The expression $a^n imes b^n$ can be simplified using the multiplication property of exponents, which states that when you multiply powers with the same base, you can add the exponents. Specifically, $a^n imes b^n$ can be rewritten as $(a imes b)^n$. This simplification is possible because the exponent $n$ is the same for both $a$ and $b$, allowing the bases to be multiplied together before raising the result to the common exponent.
• Describe the importance of the exponent being the same for both $a$ and $b$ in the expression $a^n imes b^n$.
  • The requirement for the exponent $n$ to be the same for both $a$ and $b$ in the expression $a^n imes b^n$ is crucial for applying the multiplication property of exponents. If the exponents were different, the expression could not be simplified using this property, as the bases would not have a common exponent to add together. Ensuring that the exponents are the same is a key step in manipulating expressions involving the multiplication of powers, as it allows for the application of the multiplication property and the resulting simplification of the expression.
• Analyze how the ability to apply the multiplication property of exponents to the expression $a^n imes b^n$ demonstrates a deeper understanding of exponents.
  • The ability to apply the multiplication property of exponents to the expression $a^n imes b^n$ demonstrates a deeper understanding of exponents and their underlying properties. By recognizing that the exponents can be added when the bases are multiplied, the student shows a mastery of the rules governing exponents and the ability to manipulate expressions involving powers. This understanding goes beyond simply memorizing formulas and allows the student to apply logical reasoning to simplify and solve problems involving exponents, which are essential skills in algebra and various mathematical contexts. The successful application of the multiplication property to $a^n imes b^n$ reflects a comprehensive grasp of exponents and their behavior.

Related terms

Exponent:

An exponent is a small number written to the right of and above a base number, indicating how many times the base is to be multiplied by itself.

Power:

A power is the result of raising a number (the base) to an exponent, which indicates how many times the base is to be multiplied by itself.

Multiplication Property of Exponents: The multiplication property of exponents states that when you multiply powers with the same base, you can add the exponents: $a^m imes a^n = a^{m+n}$.