Problem 60 - Algebra 1: Concepts and Skills

Question

Write the expression in exponential form. \(3 x \cdot 3 x \cdot 3 x \cdot 3 x \cdot 3 x\)

Step-by-Step Solution

Answer

The given expression in exponential form is \((3x)^5\).

Step 1: Identifying the Base and the Number of Times It Appears

In the expression, the base is \(3x\), and it appears 5 times.

Step 2: Writing in Exponential Form

An exponential expression is written as \(base^{exponent}\). Here, the base is \(3x\) and the exponent is 5, as it is being multiplied five times.

Key Concepts

Understanding Base and Exponent

The concept of base and exponent is foundational in understanding exponential expressions. In an exponential expression, the base is the number or algebraic expression that is being multiplied by itself multiple times. The exponent tells us how many times to use the base as a factor.

In the given problem, the base is represented by the expression \(3x\). This means that \(3x\) is the factor that gets multiplied over and over. The exponent, in this case, is 5. So, when you see \((3x)^5\), it simply means \(3x \cdot 3x \cdot 3x \cdot 3x \cdot 3x\), where \(3x\) is used as a factor five times.

• The base part of an exponential expression tells us what number or expression is being used repeatedly.
• The exponent lets us know the count of these repetitions.

Understanding this concept is key to accurately writing and interpreting exponential expressions.

The Role of Multiplication in Exponential Expressions

Multiplication plays a central role in forming and simplifying exponential expressions. When we multiply the same number or expression by itself repeatedly, we can use an exponential form to simplify the process.

For instance, in the problem \(3x \cdot 3x \cdot 3x \cdot 3x \cdot 3x\), instead of writing the expression with repeated multiplication, we use multiplication in the form of an exponent for simplicity. This expression can be written as \((3x)^5\).

Using exponents simplifies calculations. Rather than writing out multiple instances of multiplication, which can become cumbersome with larger numbers or complex expressions, exponents provide a shorthand way of communicating that multiplication is happening.

• Exponents reduce the labor of writing long strings of multiplication.
• They make expressions easier to read and understand.

Learning how multiplication is involved in forming and understanding exponential expressions aids students in algebra and beyond.

Identifying and Working with Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operations and can sometimes include exponential terms. Identifying the parts of an algebraic expression is crucial for writing them in different forms and solving equations.

In the expression from the exercise, \(3x\) is an algebraic expression. Here, the '3' is a coefficient multiplying the variable 'x'. When written in exponential form as \((3x)^5\), both the numerical coefficient and variable are part of the repeating base.

• Understanding coefficients and variables helps in manipulating algebraic expressions.
• It's important to recognize how each part operates when rewriting in a different form, like exponential.

This understanding is fundamental for tackling more complex algebraic problems, ensuring a solid foundation is built on how algebraic expressions function and interact.