$x^a$

Definition

$x^a$ is a mathematical expression that represents the power function, where $x$ is the base and $a$ is the exponent. This term is crucial in the context of understanding properties of exponents and scientific notation, as it forms the foundation for manipulating and working with exponential expressions.

5 Must Know Facts For Your Next Test

1. The expression $x^a$ can be read as '$x$ raised to the power of $a$' or '$x$ to the $a^{th}$ power'.
2. When the exponent $a$ is a positive integer, $x^a$ represents the product of $a$ factors of $x$, i.e., $x \times x \times x \times ... \times x$ ($a$ times).
3. If the exponent $a$ is a negative integer, $x^a$ represents the reciprocal of $x$ raised to the positive power of $|a|$, i.e., $1/x^{|a|}$.
4. The properties of exponents, such as the product rule, quotient rule, and power rule, allow for efficient manipulation of expressions involving $x^a$.
5. Scientific notation uses the form $x^a$ to represent very large or very small numbers in a compact and standardized way.

Review Questions

• Explain the meaning and interpretation of the expression $x^a$, and how it relates to the concept of a power.
  • The expression $x^a$ represents the power function, where $x$ is the base and $a$ is the exponent. This expression can be interpreted as the product of $a$ factors of $x$, i.e., $x \times x \times x \times ... \times x$ ($a$ times). The exponent $a$ indicates the number of times the base $x$ is multiplied by itself, and the resulting value is the power of $x$. This concept is fundamental to understanding properties of exponents and how to work with exponential expressions.
• Describe how the properties of exponents, such as the product rule, quotient rule, and power rule, can be used to manipulate expressions involving $x^a$.
  • The properties of exponents allow for efficient manipulation of expressions involving $x^a$. The product rule states that $x^a \times x^b = x^{a+b}$, the quotient rule states that $x^a \div x^b = x^{a-b}$, and the power rule states that $(x^a)^b = x^{a \times b}$. These rules can be applied to simplify, expand, or rewrite expressions containing $x^a$ in various ways, which is crucial for working with exponential expressions and scientific notation.
• Explain how the expression $x^a$ is used in the context of scientific notation, and discuss the importance of this representation for expressing very large or very small numbers.
  • In scientific notation, the expression $x^a$ is used to represent the power of 10 that a number is multiplied by. The number $x$ is a value between 1 and 10, and the exponent $a$ indicates the number of places the decimal point needs to be shifted to the right (for positive exponents) or left (for negative exponents) to express the original number. This compact representation of very large or very small numbers using $x^a$ is essential for working with scientific data, performing calculations, and communicating numerical information in a standardized and efficient manner.

Related terms

Exponent: The exponent is the small number written to the right of and above the base, indicating the number of times the base is to be multiplied by itself.

Power: The power is the result of raising a base to an exponent, representing repeated multiplication of the base.

Scientific Notation: Scientific notation is a way of expressing very large or very small numbers as the product of a number between 1 and 10 and a power of 10.