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Chapter 1: Problem 14

Write using exponential notation. \(6 \cdot 6 \cdot 2 \cdot 9 \cdot 9 \cdot 9 \cdot 9\)

Short Answer

Expert verified
The expression is \(6^2 \cdot 2 \cdot 9^4\).

Step by step solution

01

Identify Repeated Numbers

Look closely at the numbers involved in the multiplication and identify which numbers are repeated. In this expression, we have 6 appearing twice and 9 appearing four times.
02

Write Repeated Numbers in Exponential Form

For each set of repeated numbers, use exponential notation to express them. Since 6 appears twice, it can be written as \(6^2\). Similarly, as 9 appears four times, it can be written as \(9^4\).
03

Combine into a Single Expression

Now, write the entire expression using exponential notation by combining the exponents identified in the previous step. The original expression \(6 \cdot 6 \cdot 2 \cdot 9 \cdot 9 \cdot 9 \cdot 9\) becomes \(6^2 \cdot 2 \cdot 9^4\).

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Key Concepts

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

Exponents
Exponents are a fundamental concept in mathematics that help simplify expressions with repeated multiplication. They are utilized to show how many times a number, known as the base, is multiplied by itself. For example, in the expression \(6^2\), 6 is the base, and the exponent 2 indicates that 6 is multiplied by itself once. It effectively transforms a long multiplication string into a more concise form.

When writing numbers in exponential notation, it helps to reduce the clutter of having many numbers written out. This form is particularly useful for simplifying expressions and performing calculations with large numbers.
Multiplication
Multiplication is one of the basic arithmetic operations, essential for understanding exponential notation. It involves finding the total of one number added to itself a certain number of times.

For example, the multiplication expression \(6 \cdot 6 \cdot 2\) can be read as \(6\) added to itself once, and the product of \(6\) and \(2\) gives \(12\). Understanding multiplication helps in grasping why exponential notation works as it is built on the idea of multiplying numbers together repeatedly.
  • It links directly to repeated addition.
  • It forms the basis for understanding exponents.
Repeated Factors
The concept of repeated factors is at the heart of exponential notation. A repeated factor simply means that a specific number, or factors, are used multiple times in a multiplication expression.

In the example \(6 \cdot 6 \cdot 2 \cdot 9 \cdot 9 \cdot 9 \cdot 9\), we identify that 6 and 9 appear multiple times. Recognizing this repetition allows us to use exponents effectively. As explained in the solution, the number \(6\) appears twice and can be expressed as \(6^2\). Similarly, the number \(9\) appears four times and is represented as \(9^4\). This not only simplifies the expression but also makes it more efficient to perform calculations with larger expressions.
  • Makes expressions compact.
  • Facilitates easier calculations.

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Most popular questions from this chapter

Use a calculator to evaluate each expression for \(x=23\) and \(y=72\). \(x^{4}-y^{2}\)

Write each phrase as a variable expression. Use \(x\) to represent "a number." Twenty decreased by twice a number

Find and correct the error when evaluating each expression for \(x=3\) and \(y=5 .\) \(\begin{aligned} x^{3}+4 y &=3^{3}+4(5) \\ &=9+20 \\ &=29 \end{aligned}\)

There were about 3 million "older" Americans (ages 65 and older in \(1900 .\) By 2020 , this number is projected to increase eighteen times. Find this projected number of "older" Americans in 2020. (Source: Administration on Aging, U.S. Census Bureau)

Write each whole number in expanded form. $$20,215$$
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