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Chapter 1: Problem 4

Explain, in words, why \(7 \cdot 7 \cdot 7 \cdot 7 \cdot 7=7^{5}\)

Short Answer

Expert verified
The expression \(7 \cdot 7 \cdot 7 \cdot 7 \cdot 7\) represents multiplying the number 7 by itself 5 times. Exponential notation, such as \(7^5\), is a shorthand for this repeated multiplication, where the base number (7) is multiplied by itself a number of times equal to the exponent (5). Thus, \(7 \cdot 7 \cdot 7 \cdot 7 \cdot 7\) and \(7^5\) both represent the same process and are equivalent expressions.

Step by step solution

01

Understand Exponents

Exponents are a shorthand notation for repeated multiplication of the same number. They are used to simplify expressions and calculations that involve multiplying a number by itself several times. In this case, the exponent is 5 and the base number is 7. The exponent tells us the number of times we should multiply the base number by itself.
02

Identify Base and Exponent

In our problem, the base number is 7 and the exponent is 5. This means that we need to multiply 7 by itself 5 times, which is shown by the expression \(7 \cdot 7 \cdot 7 \cdot 7 \cdot 7\).
03

Apply the Exponent Rule

Knowing that the exponent 5 tells us we should multiply the base number 7 by itself 5 times, we can rewrite the expression \(7 \cdot 7 \cdot 7 \cdot 7 \cdot 7\) in exponential form using the notation \(7^5\). This means the same thing as multiplying 7 by itself 5 times.
04

Explain the Equality of the Two Expressions

Now that we have understood the concept of exponents and applied the exponent rule, we can explain why \(7 \cdot 7 \cdot 7 \cdot 7 \cdot 7\) is equal to \(7^5\). Both expressions represent the process of multiplying the base number 7 by itself for a total of 5 times. By using the exponent notation, we simplify this expression into \(7^5\), which is easier to read and understand. Therefore, \(7 \cdot 7 \cdot 7 \cdot 7 \cdot 7\) is equivalent to \(7^5\).

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Key Concepts

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

Base and Exponent
In mathematics, exponents are a powerful way to express the process of repeated multiplication. Understanding exponents starts with two key components: the base and the exponent.
  • Base: The base is the number that is being multiplied.
  • Exponent: The exponent tells you how many times the base is used as a factor.
Given the example \(7^5\), the number 7 is called the "base," and the number 5 is the "exponent." This tells us to multiply 7 by itself 5 times. The base serves as the repeated factor in the multiplication process, while the exponent indicates the total number of times this repetition occurs.
Exponential Notation
Exponential notation is a convenient way to express very large or repeated multiplications in a compact form. Instead of writing out all the multiplications fully, you can use this shorthand method.The exponential notation \(7^5\) translates to the expression "7 multiplied by itself 5 times." It's written as a compact expression rather than as a long multiplication sequence.
  • Exponential notation makes equations easier to read and manage.
  • It is widely used in scientific calculations due to its efficiency in expressing large numbers.
Understanding exponential notation allows us to quickly grasp complex mathematical operations, helping to simplify and solve problems more efficiently.
Repeated Multiplication
When dealing with exponents, repeated multiplication is the fundamental operation being simplified. The idea is to take a single number and multiply it by itself several times, as indicated by the exponent.For example, in the expression \(7^5\), you're performing the operation:
  • First multiply: \(7 \times 7 = 49\)
  • Second multiply: \(49 \times 7 = 343\)
  • Third multiply: \(343 \times 7 = 2401\)
  • Fourth multiply: \(2401 \times 7 = 16807\)
This method demonstrates how exponential expressions simplify repeated multiplication into a more manageable form.

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