Problem 22 - Elementary and Intermediate Algebra

Question

Write each expression in an equivalent form using an exponent. $$ -5 \cdot u \cdot u \cdot u $$

Step-by-Step Solution

Answer

The expression is equivalent to \(-5u^3\).

Step 1: Understand the Expression

In the given problem, we have an expression \[-5 \cdot u \cdot u \cdot u. \] We need to represent it using exponents. The expression contains the variable \(u\) repeated as a factor.

Step 2: Identify Repeated Factors

Observe that the variable \(u\) is used as a factor three times in the expression. This repetition suggests that we can use exponents to simplify the expression.

Step 3: Rewrite Using Exponents

To write the expression using exponents, count the number of times \(u\) appears as a factor. Here, \(u\) appears three times, so we represent it as \(u^3\).Thus, the expression becomes \[-5 \cdot u^3.\]

Step 4: Combine Terms

Combine the constant factor and the exponent expression to have the final equivalent expression: \(-5u^3.\)

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations (such as addition, subtraction, multiplication, and division). These expressions help convey various mathematical relationships and rules in a compact form. In the given exercise, the expression \(-5 \cdot u \cdot u \cdot u\) is an algebraic expression because it includes both a constant (-5) and the variable \(u\).

Algebraic expressions can be understood by looking at their components:

• **Constants:** These are fixed numbers, like -5 in our exercise.
• **Variables:** These are symbols (like u) that can represent different values. They allow expressions to be generalized.
• **Operations:** Actions like multiplication or addition that combine the constants and variables.

Understanding how these elements work together is crucial for simplifying and manipulating algebraic expressions effectively. This forms the basis of many other concepts in algebra.

Power of a Variable

In algebra, the "power of a variable" refers to how many times a variable is multiplied by itself. This is represented using exponents. For instance, in the expression given, the variable \(u\) is multiplied by itself three times. When a variable is used as a factor like this, we use an exponent to write it in a more concise way.

For example:

• When \(u\) is multiplied by itself once (e.g., \(u \times u\)), it's written as \(u^2\).
• If it is multiplied three times (e.g., \(u \times u \times u\)), it's written as \(u^3\).

The exponent tells you how many times to multiply the variable by itself. This process makes it easier to read and work with the expression. Instead of writing out long chains of multiplications, we use exponents to save time and space.

Simplifying Expressions

Simplifying expressions in algebra involves reducing them to their most basic form. This often involves using rules of arithmetic and algebra, such as combining like terms and applying exponents.

To simplify the given expression \(-5 \cdot u \cdot u \cdot u\), we:

• Identify the repeated variable, \(u\), being multiplied by itself.
• Rewrite the repeated multiplication using an exponent, resulting in \(u^3\).
• Combine the constant term (-5) with the simplified variable term \(u^3\), to form \(-5u^3\).

Simplification ensures that expressions are neat and manageable, making them easier to interpret and solve in equations. It is an essential skill for solving algebraic problems efficiently.