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Chapter 5: Problem 2

Write each expression using exponential form. $$ (-3 y)(-3 y)(-3 y) $$

Short Answer

Expert verified
The expression is \((-3y)^3\).

Step by step solution

01

Identify the base

To write the expression in exponential form, you first need to identify the repeated factor(s) in the multiplication. Here, the factor \(-3y\) is being multiplied by itself.
02

Count the number of repetitions

Count how many times the base is repeated in the multiplication. The expression \((-3y)(-3y)(-3y)\) shows that the base \(-3y\) is repeated 3 times.
03

Write using exponential notation

Using the base and the number of repetitions found, you can write the expression in exponential form as \((-3y)^3\). This states that \(-3y\) is multiplied by itself 3 times.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Base Identification
When dealing with exponential notation, the first step is to identify the base in the expression. The base is the number or expression that is multiplied by itself a certain number of times. In our example expression, \((-3y)(-3y)(-3y)\), the repeated factor is \(-3y\). This combination of a number and a variable represents the entire base of the exponential expression.
Identifying the base is crucial because it tells us what is being multiplied repeatedly. In larger mathematical problems, there may be more complex bases, but the principle remains the same: look for the factor that recurs across the multiplication.
Here are some simple points to remember when identifying bases:
  • Recognize the entire expression that repeats.
  • Bases can be numbers, variables, or a mix of both.
  • The base is what's within the parentheses often, when expressions are clearly marked out.
Multiplication Repetition
Once the base is identified, the next step is understanding how many times the base is repeated. This process involves counting the number of times the base appears in your expression.
In our example, \((-3y)(-3y)(-3y)\), the base \(-3y\) appears three times.
Counting repetition provides the exponent, which is a key part of the exponential expression.
The repetition count can instantly inform you of the power to which the base should be raised. Ensuring accuracy in counting is critical because it directly affects the overall expression.
  • Repetition informs the exponent.
  • Ensure an accurate count for precise calculations.
  • Even large numbers need simple counting to identify repetition, just like smaller figures.
Exponential Expression
Once you've identified the base and counted how many times it appears, you can rewrite the expression in exponential form. The exponential expression captures the base and the number of times it's used, both in a compact notation.
In our exercise, rewriting \((-3y)(-3y)(-3y)\) leads to \((-3y)^3\). This form indicates that the base \(-3y\) is used as a factor three separate times.
Exponential expressions simplify multiplication into a concise format, making calculations easier to manage. They also clarify the scale of repeated multiplication, which is essential for more complex equations.
  • The exponent shows the number of multiplicative instances.
  • Writing in exponential form reduces complexity.
  • It's particularly useful for simplifying algebraic expressions.

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