$x^3$

Definition

$x^3$ is a mathematical expression representing the cube of a variable $x$. It is a special case of exponents, where the exponent is 3, indicating that the variable $x$ is multiplied by itself three times. This term is crucial in the context of understanding the multiplication properties of exponents, as it demonstrates how exponents can be used to simplify and manipulate algebraic expressions.

5 Must Know Facts For Your Next Test

1. The expression $x^3$ can be expanded as $x \times x \times x$, where $x$ is multiplied by itself three times.
2. The value of $x^3$ depends on the value of $x$. For example, if $x = 2$, then $x^3 = 2 \times 2 \times 2 = 8$.
3. The multiplication properties of exponents state that $x^m \times x^n = x^{m+n}$. This means that $x^3 \times x^2 = x^{3+2} = x^5$.
4. The cube root of $x^3$ is $x$, as the cube root undoes the cubing operation. In other words, $\sqrt[3]{x^3} = x$.
5. The cube of a negative number is negative, while the cube of a positive number is positive. For example, $(-2)^3 = -8$ and $(3)^3 = 27$.

Review Questions

• Explain how the expression $x^3$ can be used to simplify algebraic expressions.
  • The expression $x^3$ can be used to simplify algebraic expressions by applying the multiplication properties of exponents. For example, if we have the expression $x^2 \times x$, we can rewrite it as $x^{2+1} = x^3$, simplifying the expression. Additionally, the cube of a variable, $x^3$, can be used to represent the product of three factors of $x$, allowing for efficient manipulation of more complex algebraic expressions.
• Describe the relationship between the cube root of $x^3$ and the value of $x$.
  • The cube root of $x^3$ is equal to $x$. This is because the cube root operation undoes the cubing operation, as $\sqrt[3]{x^3} = x$. This relationship is important in understanding the properties of exponents and how they can be used to simplify and solve algebraic equations. For example, if we have the equation $x^3 = 27$, we can find the value of $x$ by taking the cube root of both sides, resulting in $x = 3$.
• Analyze the sign of the result when raising a negative number to the power of 3.
  • When raising a negative number to the power of 3, the result will be negative. This is because the cube of a negative number is negative. For example, $(-2)^3 = -8$. This property of cubes is important to understand when working with algebraic expressions involving negative variables or numbers. The sign of the result when raising a negative number to an odd power, such as 3, will always be negative, while the sign of the result when raising a negative number to an even power will be positive.

Related terms

Exponent: An exponent is a number that indicates how many times a base number is to be multiplied by itself. In the expression $x^3$, the exponent is 3, which means $x$ is multiplied by itself three times.

Base: The base is the number or variable that is being raised to a power. In the expression $x^3$, the base is $x$.

Power: The power is the exponent, which represents the number of times the base is multiplied by itself. In the expression $x^3$, the power is 3.