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Chapter 6: Problem 1

Rewrite each expression with exponents. a. \((7)(7)(7)(7)(7)(7)(7)(7)\) b. \((3)(3)(3)(3)(5)(5)(5)(5)(5)\) c. \((1+0.12)(1+0.12)(1+0.12)(1+0.12)\)

Short Answer

Expert verified
a. \(7^8\); b. \(3^4 \times 5^5\); c. \((1+0.12)^4\).

Step by step solution

01

Identify the Base

For each expression, identify the numbers being multiplied repeatedly. These numbers will form the base for the exponent.
02

Count the Repetition

Count how many times each base number appears in its respective expression. This count will become the exponent.
03

Rewrite with Exponents for Part (a)

For part (a), notice that 7 is repeated 8 times. Thus, we rewrite it as \( 7^8 \).
04

Rewrite with Exponents for Part (b)

For part (b), notice that 3 is repeated 4 times and 5 is repeated 5 times. Thus, we rewrite this expression as \( 3^4 \times 5^5 \).
05

Rewrite with Exponents for Part (c)

For part (c), the term \((1 + 0.12)\) is repeated 4 times. Thus, we rewrite this expression as \((1 + 0.12)^4\).

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Key Concepts

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

Understanding the Base Number
In mathematics, especially when dealing with exponents, the concept of a "base number" is key. The base is the number that is going to be raised to a power. In simpler terms, it's the "main" number that gets multiplied by itself a certain number of times. For example, in the expression \[7^8,\] the "7" is the base number.
  • The base number in expressions with exponents is crucial because it tells us what number we'll multiply repeatedly.
  • In the expression \(3^4 \times 5^5,\) the bases are 3 and 5 respectively.
  • Even more interesting are expressions like \((1 + 0.12)^4.\) Here, \((1 + 0.12)\) acts as the entire base, showing that expressions can also serve as bases.
By recognizing the base, we can more clearly understand how exponentiation works.
Unpacking Repeated Multiplication
Repeated multiplication essentially refers to multiplying the same number by itself multiple times. It is the repetition of multiplying the same base number, which forms the foundation for understanding exponents. Let's take a closer look:
  • If we multiply "7" eight times: \((7)(7)(7)(7)(7)(7)(7)(7),\) we are performing repeated multiplication and can express this as \(7^8.\)
  • In mathematical expressions such as \((3)(3)(3)(3)(5)(5)(5)(5)(5),\) this can be broken down into \(3^4 \times 5^5,\) showing two different base numbers being repeatedly multiplied.
  • Likewise, \((1 + 0.12)(1 + 0.12)(1 + 0.12)(1 + 0.12)\) illustrates how repeated multiplication works with an expression as the base, resulting in \((1 + 0.12)^4.\)
Recognizing repeated patterns is instrumental in simplifying expressions using exponents.
Crafting an Expression with Exponents
An expression with exponents is a compact way to write repeated multiplication. By employing exponents, you replace long strings of multiplication with a simpler notation that describes the extent of the repetition. Have a look at these examples:
  • \((7)(7)(7)(7)(7)(7)(7)(7)\) transforms into \(7^8.\) Here, "8" tells you how many times "7" is used in the multiplication.
  • \((3)(3)(3)(3)(5)(5)(5)(5)(5)\) is written as \(3^4 \times 5^5,\) clearly indicating that "3" is repeated 4 times and "5" is repeated 5 times.
  • In expressions like \((1 + 0.12)(1 + 0.12)(1 + 0.12)(1 + 0.12),\) you write \((1 + 0.12)^4,\) showing that the whole expression is repeated 4 times.
Using exponents simplifies reading and writing mathematical expressions, creating a clearer pathway to calculation.

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Most popular questions from this chapter

Write each number in exponential form with base 3 . a. 81 b. 27 c. 9 d. \(\frac{1}{3}\) e. \(\frac{1}{9}\) f. 1

Mini-Investigation In this exercise you will explore the equation \(y=10(1-0.25)^{x}\). a. Find \(y\) for some large positive values of \(x\), such as 100,500 , and 1000 . What happens to \(y\) as \(x\) gets larger and larger? b. The calculator will say \(y\) is 0 when \(x\) equals 10,000 . Is this correct? Explain why or why not. c. Find \(y\) for some large negative values of \(x\), such as \(-100,-500\), and \(-1000\). What happens to \(y\) as \(x\) moves farther and farther from 0 in the negative direction?

Write each number in scientific notation. a. \(34,000,000,000\) (a) b. \(-2,100,000\) c. 10,060

Write an equivalent expression in the form \(a \cdot b^{n}\). ( \(\hbar\) ) a. \(3 x \cdot 5 x^{3}\) b. \(x \cdot x^{5}\) c. \(2 x^{3} \cdot 2 x^{3}\) d. \(3.5(x+0.15)^{4} \cdot(x+0.15)^{2}\) e. \(\left(2 x^{3}\right)^{3}\) f. \(\left[3(x+0.05)^{3}\right]^{2}\)

APPLICATION Webster owns a set of antique dining-room furniture that has been in his family for many years. The historical society tells him that furniture similar to his has been appreciating in value at \(10 \%\) per year for the last 20 years and that his furniture could be worth \(\$ 10,000\) now. a. Which letter in the equation \(y=A(1+r)^{x}\) could represent the value of the furniture 20 years ago when it started appreciating? (a) b. Substitute the other given information into the equation \(y=A(1+r)^{x}\). (A) c. Solve your equation in 4 b to find how much Webster's furniture was worth
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