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Chapter 6: Problem 1

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

Short Answer

Expert verified
81 is \(3^4\), 27 is \(3^3\), 9 is \(3^2\), \(\frac{1}{3}\) is \(3^{-1}\), \(\frac{1}{9}\) is \(3^{-2}\), and 1 is \(3^0\).

Step by step solution

01

Recognize the Base

We need to express each number in the form of powers of 3. This means if we have a number like 81, we should be thinking about what power of 3 results in 81.
02

Express 81 as a Power of 3

81 can be expressed as a power of 3. We need to calculate how many 3s are multiplied to get 81.\[81 = 3^4\]So, 81 is \(3^4\).
03

Express 27 as a Power of 3

27 is another number we need to express with base 3.\[27 = 3^3\]Therefore, 27 is \(3^3\).
04

Express 9 as a Power of 3

Next, we determine how to express 9 in exponential form with base 3.\[9 = 3^2\]Thus, 9 is \(3^2\).
05

Express \(\frac{1}{3}\) as a Power of 3

We need to write \(\frac{1}{3}\) as a power of 3. Recognize that \(\frac{1}{3}\) is the reciprocal of 3.\[\frac{1}{3} = 3^{-1}\]Hence, \(\frac{1}{3}\) is \(3^{-1}\).
06

Express \(\frac{1}{9}\) as a Power of 3

Now, let's express \(\frac{1}{9}\) as a power of 3. Since \(9 = 3^2\), its reciprocal is:\[\frac{1}{9} = 3^{-2}\]So, \(\frac{1}{9}\) is \(3^{-2}\).
07

Express 1 as a Power of 3

Finally, any number raised to the power of 0 is 1.\[1 = 3^0\]Therefore, 1 is \(3^0\).

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Key Concepts

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

Powers of Numbers
Exponents are a simple yet powerful tool in mathematics. When we talk about the power of a number, we're referring to how many times a number (known as the base) is multiplied by itself. For example, when we say 3 raised to the power of 4, written as \(3^4\), we mean \(3 \times 3 \times 3 \times 3\). This repeated multiplication is what gives us exponential growth, a concept central to many math applications.
  • Exponential form is helpful for simplifying expressions and calculations.
  • They are written as \(b^n\), where \(b\) is the base and \(n\) is the exponent.
  • Exponents show how many times the base is used as a factor.
Recognizing and utilizing powers of numbers gives you the ability to break down complex numbers into more manageable parts, making calculations easier.
Base 3
Numbers can be expressed using a variety of bases; however, base 3 is particularly interesting. In base 3, each number is represented as a power of 3. Let鈥檚 explore this concept further. For example, the number 81 is \(3^4\) when written in base 3. This means multiplying four 3s together: \(3 \times 3 \times 3 \times 3 = 81\).
  • Base 3 is just one way to express numbers exponentially.
  • It is similar to other bases like base 10 or base 2; the base number is the one that is repeated as a factor.
  • Common examples include 81 as \(3^4\), 27 as \(3^3\), 9 as \(3^2\), and so on.
Understanding how to express numbers in base 3 enriches your mathematical toolset and enhances problem-solving skills.
Reciprocal
A reciprocal is simply the inverse of a number. If you have a number and multiply it with its reciprocal, you get 1. For example, the reciprocal of 3 is \(\frac{1}{3}\). In the context of exponents, taking the reciprocal is similar to changing the sign of the exponent.
  • Reciprocal of \(n\) is \(\frac{1}{n}\).
  • Multiplying a number by its reciprocal results in 1.
  • In exponents, the reciprocal changes the exponent from positive to negative.
Converting numbers to their reciprocals in exponential form broadens your understanding of how exponents work, particularly with division and fractions.
Negative Exponents
Negative exponents indicate the reciprocal of a number raised to a positive exponent. For instance, \(3^{-1}\) represents \(\frac{1}{3}\). Instead of multiplying, you are essentially dividing 1 by the base raised to the positive equivalent of the negative exponent.
  • A negative exponent represents a reciprocal.
  • \(b^{-n}\) is equivalent to \(\frac{1}{b^n}\).
  • It simplifies expressions by allowing division to be represented alongside multiplication.
Mastering negative exponents allows for elegant solutions to complex problems and provides a deeper understanding of how numbers interact through division and multiplication in linear algebra.

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Most popular questions from this chapter

Draw a "starting" line segment \(2 \mathrm{~cm}\) long on a sheet of paper. a. Draw a segment 3 times as long as the starting segment. How long is this segment? b. Draw a segment 3 times as long as the segment in 10a. How long is this segment? c. Use the starting length and an exponent to write an expression that gives the length in centimeters of the next segment you would draw. (a) d. Use the starting length and an exponent to write an expression that gives the length in centimeters of the longest segment you could draw on a \(100 \mathrm{~m}\) soccer field.

Earth is \(1.5 \times 10^{11} \mathrm{~m}\) from the Sun. Light travels at a speed of \(3 \times 10^{8} \mathrm{~m} / \mathrm{s}\). How long does it take light to travel from the Sun to Earth? Answer to the nearest minute.

Eliminate factors equivalent to 1 and rewrite the right side of this equation. $$ \frac{x^{5} y^{4}}{x^{2} y^{3}}=\frac{x+x+x+x \cdot x+y+y+y+y}{x+x+y+y+y} $$

Write each number in scientific notation. How does your calculator show each answer? a. 250 b. \(7,420,000,000,000\) c. \(-18\)

Grace manages a local charity. A wealthy benefactor has offered two options for making a donation over the next year. One option is to give \(\$ 50\) now and \(\$ 25\) each month after that. The second option is to give \(\$ 1\) now and twice that amount next month; each month afterward the benefactor would give twice the amount given the month before. a. Determine how much Grace's charity would receive each month under each option. Use a table to show the values over the course of one year. (in) b. Use another table to record the total amount Grace's charity will have received after each month. c. Let \(x\) represent the number of the month (1 to 12), and let \(y\) represent the total amount Grace received after each month. On the same coordinate axes, graph the data for both options. How do the graphs compare? d. Which option should Grace choose? Why?
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