Problem 21 - Elementary and Intermediate Algebra

Question

Write each expression in an equivalent form using an exponent. $$ -4 \cdot t \cdot t \cdot t \cdot t \cdot t $$

Step-by-Step Solution

Answer

The expression is equivalent to \(-4t^5\).

Step 1: Identify the Base and Repeated Multiplication

In the expression, we have the base, which is \( t \), and it is multiplied by itself several times. Our goal is to express this repeated multiplication using an exponent.

Step 2: Count the Repetitions of the Base

Count how many times the base \( t \) is multiplied. In the expression \(-4 \cdot t \cdot t \cdot t \cdot t \cdot t\), \( t \) appears 5 times.

Step 3: Rewrite as an Exponent

Since \( t \) is repeated 5 times, this can be written as \( t^5 \). Therefore, the expression with an exponent becomes \(-4t^5\). This shows that the power of 5 applies only to \( t \), not to the coefficient \(-4\).

Step 4: Combine Elements for Final Expression

The coefficient \(-4\) remains in front and is multiplied by the power of \( t \). Therefore, the complete expression with the exponent is \(-4t^5\).

Key Concepts

Base and Exponent

Understanding the concepts of base and exponent is crucial when dealing with powers and exponential notation in algebra. The base is the value that is being multiplied repeatedly, and the exponent denotes how many times the base is multiplied by itself. In the expression

• \( t^5 \)

- 't' is the base. It represents the number that you are repeatedly multiplying.- '5' is the exponent, which tells us that 't' is multiplied by itself 5 times.

The process of taking something that is repeatedly multiplied and converting it into exponential form simplifies expressions and makes calculations easier. For example, instead of writing \( t \cdot t \cdot t \cdot t \cdot t \), it's much neater and efficient to write \( t^5 \). This understanding helps in solving complex algebraic expressions and in performing calculations more effectively.

Multiplication of Variables

When working with variables, particularly in algebraic expressions, you'll often perform multiplication. Variables are symbols, like 't', that represent values which can change. In algebra, the multiplication of variables can be expressed in compact forms using exponents.

For instance, if we take the expression

• \( t \cdot t \cdot t \cdot t \cdot t \),

this means 'multiply t five times', which is visually bulky. By writing it as \( t^5 \), we use concise exponential notation indicating the same operation.

Remember, when multiplying variables such as 't' several times, the resulting product is expressed with the base being the variable, and the exponent showing the count of how often it appears in the multiplication array. This process isn't about changing values but about representing the same quantity in a simpler and more manageable form.

Algebraic Expressions

Algebraic expressions are mathematical phrases that comprise numbers, variables, and operations. They are fundamental in algebra as they allow us to express mathematical ideas precisely and concisely.

• A typical algebraic expression might look like
• \(-4t^5\),
• which consists of a numerical coefficient and a variable part with an exponent.

The expression can be broken down into its components: - **Coefficient:** Here, -4 is the numerical part that multiplies with the variable.- **Variable:** 't' is the placeholder that can hold different values. - **Exponent:** The '5' shows how many times the base 't' is used in the multiplication.

An algebraic expression combines these elements to represent complex mathematical relationships in a very compact form, enabling us to solve equations, graph functions, and explore mathematical patterns. Understanding how to break down and manipulate these expressions is central to mastering algebra.