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Chapter 1: Problem 7

Write using exponential notation. \(6 \cdot 6 \cdot 5 \cdot 5 \cdot 5\)

Short Answer

Expert verified
\(6^2 \times 5^3\)

Step by step solution

01

Identify Repeated Factors

Look at the expression given: \(6 \cdot 6 \cdot 5 \cdot 5 \cdot 5\). Identify the numbers that are repeated. Here, the number 6 is repeated twice and the number 5 is repeated three times.
02

Write Each Group in Exponential Notation

Write each group of repeated factors using exponents. The number 6, repeated twice, is written as \(6^2\). The number 5, repeated three times, is written as \(5^3\).
03

Express the Complete Expression with Exponents

Combine the exponential notations into a single expression: \(6^2 \times 5^3\). This represents the product of all the counted terms in exponential form.

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Key Concepts

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

Repeated Factors
When we talk about repeated factors, we're referring to numbers that appear more than once in a multiplication sequence. In the expression \(6 \cdot 6 \cdot 5 \cdot 5 \cdot 5\), some numbers are repeated, specifically the numbers 6 and 5. There are a few key points about repeated factors:
  • Repeated factors simplify calculations. Instead of multiplying a number by itself multiple times, we use shortcuts.
  • They establish patterns which help in understanding larger algebraic expressions.
Identifying these patterns is crucial. It sets the stage for simplifying expressions further by using exponential notation.
Writing with Exponents
Exponents provide a way to express repeated multiplication of the same number. Instead of writing a number like 6 two times in a multiplication, we can use exponents as a shorthand. In our example, we have the factor 6 repeated twice: \(6 \cdot 6\). This is written with an exponent as \(6^2\). Similarly, the number 5 is repeated three times: \(5 \cdot 5 \cdot 5\), and is noted as \(5^3\).Expanding on exponents:
  • The base (bottom number in \(6^2\)) shows the "repeated" number.
  • The exponent (top number in \(6^2\)) tells how many times it's repeated.
This helps visualize and manage more complex problems by breaking them down into their simplest parts.
Prealgebra Concepts
Prealgebra sets the foundation for understanding how numbers work before delving into more complex mathematics. It introduces concepts like exponents, among others.Understanding exponents is key because:
  • They help simplify calculations, making large numbers easier to handle.
  • They're foundational for learning algebra, as they often appear in equations and graphing.
With the exercise of rewriting \(6 \cdot 6 \cdot 5 \cdot 5 \cdot 5\) using exponents to become \(6^2 \times 5^3\), students learn how patterns help solve math problems efficiently. This approach is essential for mastering higher levels of math.

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Most popular questions from this chapter

Check each addition below. If it is incorrect, find the correct answer. $$ \begin{array}{r} 14 \\ 173 \\ 86 \\ +257 \\ \hline 520 \end{array} $$

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