The expression $(2x)^3$ represents the cube of the expression $2x$. It is an example of a power of a power, where the base is the expression $2x$ and the exponent is 3.
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The expression $(2x)^3$ can be simplified to $8x^3$, using the power of a power rule.
The exponent 3 in $(2x)^3$ indicates that the expression $2x$ is multiplied by itself three times.
Raising a power to a power is a common operation in algebra and is used to simplify and manipulate expressions with exponents.
The coefficient 2 in the expression $2x$ is also raised to the power of 3, resulting in a coefficient of 8 in the simplified expression $8x^3$.
Understanding the properties of exponents, including the power of a power rule, is essential for simplifying and evaluating expressions with exponents.
Review Questions
Explain the step-by-step process of simplifying the expression $(2x)^3$.
To simplify the expression $(2x)^3$, we can apply the power of a power rule. The base is $2x$, and the exponent is 3. Using the rule $(a^m)^n = a^{mn}$, we can rewrite $(2x)^3$ as $2^3 \cdot x^3$, which simplifies to $8x^3$. The key steps are: 1) Identify the base and the exponent, 2) Apply the power of a power rule, and 3) Simplify the resulting expression.
Describe how the value of the expression $(2x)^3$ changes as the value of $x$ increases or decreases.
The expression $(2x)^3$ is a polynomial function, where the value of the expression depends on the value of $x$. As the value of $x$ increases, the value of $(2x)^3$ will increase at a faster rate due to the exponent of 3. Conversely, as the value of $x$ decreases, the value of $(2x)^3$ will decrease at a faster rate. This is because the exponent of 3 amplifies the effect of changes in the value of $x$, resulting in a nonlinear relationship between $x$ and the expression $(2x)^3$.
Analyze the relationship between the expression $(2x)^3$ and the expression $8x^3$, and explain how they are connected.
The expressions $(2x)^3$ and $8x^3$ are equivalent and represent the same mathematical concept. The expression $(2x)^3$ can be simplified to $8x^3$ by applying the power of a power rule. The key connection is that the coefficient of 2 in the base $2x$ is raised to the power of 3, resulting in a coefficient of 8 in the simplified expression $8x^3$. This demonstrates how raising a power to a power can be used to simplify and manipulate expressions with exponents, which is an essential skill in algebra and mathematics.
Related terms
Power of a Power: The rule for raising a power to a power, where $(a^m)^n = a^{mn}$.
Exponent: The exponent represents the number of times the base is multiplied by itself.
Simplifying Exponents: The process of rewriting an expression with exponents in a more compact form.