Problem 62 - Intermediate Algebra

Question

Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen. a. double the cube of a number b. the cube of double a number

Step-by-Step Solution

Answer

a. \( 2x^3 \); b. \( 8x^3 \).

Step 1: Understand the Phrase 'double the cube of a number'

In this phrase, we first need to "cube a number" which can be represented by the variable \( x^3 \) if we let the number be \( x \). Then, "double the cube of a number" means multiplying this result by 2, giving us the expression \( 2x^3 \).

Step 2: Understand the Phrase 'the cube of double a number'

This phrase requires us to first "double a number," which can be represented as \( 2x \) if we let the number be \( x \). Next, "the cube of" this result involves raising it to the power of 3, resulting in the expression \( (2x)^3 \).

Step 3: Simplify the Expression '(2x)^3'

The expression \( (2x)^3 \) involves expanding the cube of a product, which is calculated as \( (2x)(2x)(2x) = 8x^3 \).

Key Concepts

Cubic Function

A cubic function is a type of polynomial that includes a variable raised to the power of three. In mathematical terms, if we have a variable \( x \), then a cubic function can be represented as \( x^3 \). These functions are essential in mathematics as they describe various real-world phenomena, such as volume calculations and certain types of growth.

• The "cube" of a number refers to multiplying that number by itself three times. For example, \( 3^3 = 3 \times 3 \times 3 = 27 \).
• In algebraic expressions, cubic functions can appear as part of more complex calculations, like within the expression \( 2x^3 \).
• Cubic functions are fundamentally different from squares (\( x^2 \)) and linear functions (\( x \)) because they involve three-dimensional concepts.

Understanding cubic functions can help students tackle more complex algebraic problems by understanding how numbers behave under multiplication. Cubic functions can describe not only the rate of change but also can alter the shape of a graph significantly.

Doubling in Algebra

Doubling in algebra involves multiplying a number or an expression by 2. This is a straightforward mathematical concept that forms a part of many algebraic expressions. It's essential to carefully identify when and where doubling is required in an expression based on the problem statement.

• The expression "double \( x \)" translates to \( 2x \), meaning twice the value of \( x \).
• In some cases, this might involve additional operations, such as cubing, involving further mathematical expressions like \( (2x)^3 \).
• Doubling is not only limited to single variables; it can apply to entire expressions, such as "doubling the sum," leading to an increased result.

Doubling should be understood as both a straightforward operation and a foundational concept that aids in constructing more complex algebraic expressions. Mastery of doubling in algebra can enhance a student’s ability to solve equations that involve repeated and scaled increases.

Variables in Algebra

Variables are symbols used to represent numbers in algebraic expressions and equations. They form the basis for much of algebra, allowing for generalization and manipulation of expressions without knowing the specific numbers involved.

• Common symbols for variables include \( x \), \( y \), and \( z \). Each represents an unknown value or a value that can change.
• In the expression \( 2x^3 \), \( x \) is a variable and plays a critical role in defining the behavior of the expression.
• Variables enable algebraic expressions to be flexible. One expression can solve numerous problems with different variable values.

Understanding how variables function is crucial for solving algebraic problems, as they can represent numbers in formulas or even entire expressions. They are the backbone of algebraic reasoning, offering a level of abstraction needed to handle more complex mathematical scenarios.