$a^n$

Definition

$a^n$ is a mathematical expression that represents the exponentiation of a base $a$ with an exponent $n$. It is a fundamental concept in algebra and is used to represent repeated multiplication of a number by itself a certain number of times.

5 Must Know Facts For Your Next Test

1. The expression $a^n$ represents the product of $n$ factors of $a$, where $a$ is the base and $n$ is the exponent.
2. When the exponent $n$ is a positive integer, $a^n$ is equal to the product of $n$ copies of the base $a$.
3. If the exponent $n$ is 0, then $a^0 = 1$, regardless of the value of the base $a$.
4. Negative exponents represent the reciprocal of the base raised to the absolute value of the exponent, i.e., $a^{-n} = \frac{1}{a^n}$.
5. Fractional exponents, such as $a^{\frac{1}{2}}$, represent the square root of the base $a$.

Review Questions

• Explain the meaning of the expression $a^n$ and how it relates to repeated multiplication.
  • The expression $a^n$ represents the exponentiation of a base $a$ with an exponent $n$. It is equivalent to the product of $n$ factors of $a$, where the base $a$ is multiplied by itself $n$ times. For example, $3^4$ is equal to $3 \times 3 \times 3 \times 3$, which is the same as multiplying the base $3$ by itself 4 times. This concept of repeated multiplication is a fundamental aspect of exponents and is used extensively in algebra and other mathematical contexts.
• Describe the properties of exponents, including the cases when the exponent is 0 or negative.
  • The properties of exponents include the following: - When the exponent $n$ is a positive integer, $a^n$ is equal to the product of $n$ copies of the base $a$. - If the exponent $n$ is 0, then $a^0 = 1$, regardless of the value of the base $a$. - Negative exponents represent the reciprocal of the base raised to the absolute value of the exponent, i.e., $a^{-n} = \frac{1}{a^n}$. - Fractional exponents, such as $a^{\frac{1}{2}}$, represent the square root of the base $a$.
• Analyze the relationship between the base, exponent, and the value of $a^n$ in the context of the Multiplication Properties of Exponents.
  • The expression $a^n$ is a fundamental component of the Multiplication Properties of Exponents, which describe how exponents behave when multiplying or dividing expressions with the same base. Specifically, the properties state that $a^m \times a^n = a^{m+n}$ and $\frac{a^m}{a^n} = a^{m-n}$. These properties demonstrate the relationship between the base $a$, the exponents $m$ and $n$, and the resulting value of the expression $a^n$. Understanding these properties is crucial for simplifying and manipulating expressions involving exponents, which is a key skill in the context of the Multiplication Properties of Exponents.

Related terms

Base:

The number or variable that is being raised to a power, represented by $a$ in the expression $a^n$.

Exponent:

The number that indicates how many times the base is multiplied by itself, represented by $n$ in the expression $a^n$.

Exponentiation:

The operation of raising a number or variable to a power, as represented by the expression $a^n$.