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Chapter 1: Problem 4

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

Short Answer

Expert verified
The expression is written as \(6^7\).

Step by step solution

01

Identify the Base Number

In exponential notation, the base is the number that is multiplied by itself. Here, the base number is 6 because it is repeated in the multiplication 7 times.
02

Count the Number of Times the Base is Used

Count how many times the number 6 appears in the multiplication. In this case, 6 is used 7 times.
03

Write in Exponential Form

In exponential notation, use the format \(a^b\), where \(a\) is the base and \(b\) is the exponent indicating how many times the base is multiplied by itself. Therefore, write \(6^7\).

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Key Concepts

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

Base Number
In exponential notation, the base number is fundamental for understanding how this mathematical shorthand works. It is the number that you multiply over and over again. Imagine it as the core number that sets the foundation. In the original exercise, the sequence of multiplying the number 6 repeatedly identifies 6 as the base number. It acts like a building block in exponential expressions, solidifying what you will be repetitively multiplying together. Every exponential expression revolves around its base number, which dictates what number is repeatedly used.
Exponent
While the base number tells you 'what number' is being multiplied, the exponent lets you know 'how many times' to multiply that base number. In exponential notation, the exponent is the small number found at the top right corner of the base number. It tells you the count of the multiplicative repetition. For example, in the expression given, if 6 is multiplied by itself seven times, then the exponent is 7. The exponent gives a glance over a long multiplication process and condenses it into a more structured form. It's like a silent little guide that instructs you to keep repeating the base number that many times.
Exponential Form
The exponential form is a streamlined way to write repeated multiplication. It's a shorthand notation where you only see two numbers: the base and the exponent. This format is particularly efficient when dealing with numerous multiplications of the same number, offering a more manageable and clear expression. Instead of writing out 6 multiplied by itself seven times, you use the exponential form - this would be written as \(6^7\). The exponential form makes mathematical expressions cleaner, tidier, and easier to understand at a glance, highlighting the power of mathematics in simplifying complex operations.
Multiplication
Repeated multiplication is at the heart of understanding exponential notation. In the given exercise, the repeated multiplication of 6 is converted into a simpler form using exponents. Initially, the expression involves writing 6 seven times with the multiplication sign in between each pair, such as \(6 \cdot 6 \cdot 6 \cdot 6 \cdot 6 \cdot 6 \cdot 6\). This repetitive process can become tedious, especially with larger numbers or more repetitions. That's where exponential notation steps in to simplify - turning long strands of multiplication into compact expressions like \(6^7\). Therefore, understanding multiplication is a gateway to embracing exponential notation.

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Most popular questions from this chapter

Check to see whether each number written in standard form matches the number written in words. If not, correct the number in words. How large is a trillion? To get an idea, one trillion seconds is over 31 thousand years. Look up "trillion" in a dictionary and use the definition to write this number in standard form.

Explain why the phrase "Subtract 7 from 10 " translates to \(" 10-7 . "\)

Find \(26 \div 5\) using the process of repeated subtraction.

In your own words, explain the commutative property of addition.

Write each whole number in expanded form. $$3470$$
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