key term - $8x^3$

Definition

$8x^3$ is a polynomial expression that represents the product of a constant, 8, and a variable, $x$, raised to the power of 3. This term is an important component in the study of multiplication properties of exponents, as it demonstrates how to apply these properties to simplify and manipulate polynomial expressions.

5 Must Know Facts For Your Next Test

1. The exponent 3 in $8x^3$ indicates that the variable $x$ is being multiplied by itself three times.
2. The constant 8 in $8x^3$ is the coefficient, which represents the numerical factor that multiplies the variable expression $x^3$.
3. When simplifying expressions with exponents, the multiplication property of exponents states that $x^a \cdot x^b = x^{a+b}$.
4. The power rule for exponents states that $(x^a)^b = x^{a \cdot b}$, which can be used to simplify expressions like $8x^3$.
5. Expressions like $8x^3$ can be further simplified by combining like terms, which involves adding the coefficients of terms with the same variable and exponent.

Review Questions

• Explain how the exponent 3 in the term $8x^3$ affects the value of the expression.
  • The exponent 3 in the term $8x^3$ indicates that the variable $x$ is being multiplied by itself three times. This means that the value of $x$ is being cubed, or raised to the power of 3. For example, if $x = 2$, then $8x^3 = 8(2^3) = 8(8) = 64. The exponent 3 has a significant impact on the overall value of the expression, as it causes the variable to increase exponentially.
• Describe how the multiplication property of exponents can be used to simplify the expression $8x^3$.
  • The multiplication property of exponents states that $x^a \cdot x^b = x^{a+b}$. In the case of $8x^3$, the variable $x$ is being raised to the power of 3. Using the multiplication property, we can simplify this expression as follows: $8x^3 = 8 \cdot x^3$. This means that the exponents can be added together, resulting in $x^3$. The constant 8 remains as the coefficient of the simplified expression.
• Analyze how the term $8x^3$ can be further simplified by combining like terms.
  • If the expression $8x^3$ is part of a larger polynomial expression, it can be further simplified by combining like terms. Like terms are terms that have the same variable and exponent. In this case, $8x^3$ would be considered a like term with any other term that has the variable $x$ raised to the power of 3. For example, if the expression was $5x^2 + 8x^3 - 2x^3$, the like terms $8x^3$ and $-2x^3$ could be combined by adding their coefficients, resulting in $6x^3$. This process of combining like terms allows for the simplification of more complex polynomial expressions.

Related terms

Polynomial:

A polynomial is an expression that consists of variables and coefficients, where the variables are raised to non-negative integer powers.

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.

Coefficient:

A coefficient is a number that is multiplied by a variable in an algebraic expression.