key term - $a^{m \times n}$

Definition

$a^{m \times n}$ is a mathematical expression that represents the result of raising the base $a$ to the power of the product of $m$ and $n$. This term is particularly relevant in the context of understanding the multiplication properties of exponents, which is a crucial concept in elementary algebra. The expression $a^{m \times n}$ can be interpreted as taking the base $a$, and raising it to the power that is the product of $m$ and $n$. This operation is often used to simplify and manipulate expressions involving exponents, which is an essential skill for solving algebraic problems.

5 Must Know Facts For Your Next Test

1. The expression $a^{m \times n}$ represents the result of raising the base $a$ to the power of the product of $m$ and $n$.
2. The power rule, $(a^m)^n = a^{m \times n}$, can be used to simplify expressions involving $a^{m \times n}$.
3. The multiplication property of exponents, $a^m \times a^n = a^{m+n}$, can be used to rewrite $a^{m \times n}$ as $a^{m+n}$ when the bases are the same.
4. The value of $a^{m \times n}$ depends on the values of $a$, $m$, and $n$, and can be used to represent a wide range of mathematical concepts, such as exponential growth and decay.
5. Understanding the properties and applications of $a^{m \times n}$ is crucial for solving various algebraic problems, including those involving exponents, powers, and exponential functions.

Review Questions

• Explain how the power rule, $(a^m)^n = a^{m \times n}$, can be used to simplify expressions involving $a^{m \times n}$.
  • The power rule, $(a^m)^n = a^{m \times n}$, can be used to simplify expressions involving $a^{m \times n}$ by rewriting the expression as a single exponent. For example, if we have the expression $(a^3)^4$, we can use the power rule to rewrite it as $a^{3 \times 4} = a^{12}$. This simplification can be useful in algebraic manipulations and problem-solving.
• Describe how the multiplication property of exponents, $a^m \times a^n = a^{m+n}$, can be used to rewrite $a^{m \times n}$ in a different form.
  • The multiplication property of exponents, $a^m \times a^n = a^{m+n}$, can be used to rewrite $a^{m \times n}$ as $a^{m+n}$ when the bases are the same. For example, if we have the expression $a^{2 \times 3}$, we can use the multiplication property to rewrite it as $a^{2} \times a^{3} = a^{2+3} = a^{5}$. This can be a useful simplification when working with expressions involving exponents.
• Analyze how the value of $a^{m \times n}$ can be used to represent and understand various mathematical concepts, such as exponential growth and decay.
  • The expression $a^{m \times n}$ can be used to represent and understand a wide range of mathematical concepts, such as exponential growth and decay. For example, in the context of exponential growth, the value of $a^{m \times n}$ can be used to model the growth of a quantity over time, where $a$ represents the growth factor, $m$ represents the number of time periods, and $n$ represents an additional factor that affects the growth rate. Similarly, in the context of exponential decay, $a^{m \times n}$ can be used to model the decline of a quantity over time, where $a$ represents the decay factor, $m$ represents the number of time periods, and $n$ represents an additional factor that affects the decay rate. Understanding the properties and applications of $a^{m \times n}$ is crucial for solving problems involving exponential functions and other mathematical models.

Related terms

Exponent:

An exponent is a number that indicates how many times the base number is to be multiplied by itself. It is represented as a small number written to the right and above the base number.

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

Power Rule: The power rule states that when you raise a power to another power, you multiply the exponents: $(a^m)^n = a^{m \times n}$.