Problem 95 - Introductory and Intermediate Algebra for College Students 4th

Question

Let \(x\) represent the number. Express each sentence as a single algebraic expression. Then simplify the expression. Cube a number. Subtract 4 from this exponential expression. Multiply this difference by 5 .

Step-by-Step Solution

Answer

The simplified expression is \(5x^3-20\)

Step 1: Cubing the number

Express a number that has been cubed with algebraic expressions as \(x^3\)

Step 2: Subtract 4

Subtract 4 from this cubic expression, yielding \(x^3-4\)

Step 3: Multiply by 5

This difference must be multiplied by 5. In algebraic expression, it would be \(5*(x^3-4)\)

Step 4: Final Simplification

Ensure all multiplications are performed and simplify the expression as \(5x^3-20\)

Key Concepts

Cubing a Number

Cubing a number is an operation in which a number is multiplied by itself three times. It's represented as an exponential expression where the base is the number being cubed and the exponent is 3. In algebraic terms, if we have a number represented by the variable x, then cubing that would be expressed as x³. This can also be seen as x × x × x.

When working with cubing, it's important to remember that every number, positive or negative, once cubed, maintains the sign — positive times positive times positive yields a positive, and negative times negative times negative also yields a negative. Cubing provides us with a powerful tool to visualize volume as well, for example, the volume of a cube with sides of length x would be x³.

Simplifying Expressions

Simplifying expressions in algebra consists of reducing the complexity of an expression without changing its value. This process includes combining like terms, distributing factors across terms within parentheses, and performing any arithmetic operations. It focuses on making the expression as easy to understand as possible. For example, when you're given the expression 5*(x³ - 4), simplifying it would include distributing the 5 to both terms inside the parentheses.

To do this effectively, one has to apply the distributive property of multiplication over subtraction. The expression is then transformed from 5*(x³ - 4) to 5x³ - 20. This resulting expression is much simpler and direct, showing clear terms that describe the relationship between the variables and constants involved.

Exponential Expressions

Exponential expressions are powerful mathematical representations that involve a base raised to a certain power or exponent. In the context of algebra, these expressions are essential for denoting repeated multiplication and can greatly simplify the way we write numbers that grow or shrink rapidly.

Understanding the Base and Exponent

For instance, the base in the expression x³, is x, and the exponent is 3, indicating that x is to be multiplied by itself two additional times. The exponent dictates the number of times the base is used as a factor in the multiplication.

Properties of Exponents

Exponents have several properties that can make simplifying expressions easier, like the power of a product rule, power of a power rule, and power of a quotient rule. For example, the power of a product rule allows us to individually exponentiate numbers inside a multiplication before multiplying them, simplifying complex expressions before working them out fully.