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Chapter 4: Problem 27

Write \(64,256,\) and \(1,024\) using integer exponents and the same base.

Short Answer

Expert verified
64 = 2^6, 256 = 2^8, 1,024 = 2^{10}.

Step by step solution

01

Identify the Base

Find the common base for the numbers 64, 256, and 1024. Since each of these numbers is a power of 2, we will use 2 as the base.
02

Express 64 as an Exponent

Express 64 as a power of 2. Since \[\begin{equation}64 = 2^6\text{,} \end{equation}\]we can write 64 as \[\begin{equation}2^6. \end{equation}\]
03

Express 256 as an Exponent

Express 256 as a power of 2. Since \[\begin{equation}256 = 2^8\text{,} \end{equation}\]we can write 256 as \[\begin{equation}2^8. \end{equation}\]
04

Express 1,024 as an Exponent

Express 1,024 as a power of 2. Since \[\begin{equation}1,024 = 2^{10}\text{,} \end{equation}\]we can write 1,024 as \[\begin{equation}2^{10}. \end{equation}\]

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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, the concepts of base and exponent are fundamental when dealing with powers and exponents. The 'base' is the number that is being multiplied, while the 'exponent' (or power) indicates how many times the base is used as a factor.
A simple example is \[\begin{equation}2^3.\end{equation}\]\
powers of 2
Understanding the powers of 2 is crucial, especially in the context of binary systems used in computer science. Powers of 2 refer to the repeated multiplication of the number 2 by itself.
Here鈥檚 a quick look at some powers of 2:
  • \[2^1 = 2\]
  • \[2^2 = 4\]
  • \[2^3 = 8\]
  • \[2^4 = 16\]
  • \[2^5 = 32\]
    • ...and so forth.
These values grow exponentially, meaning each step up in the exponent doubles the previous value. As you can see, it becomes larger quite quickly!
expressing numbers as exponents
Expressing numbers as exponents is a way of simplifying and working with larger numbers. In our exercise, we expressed 64, 256, and 1,024 as powers of 2. When we say we are expressing a number as an exponent, we are writing it in the form of \[\begin{equation}b^n ,\end{equation}\]\

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Most popular questions from this chapter

Evan made this toothpick pattern. He described the pattern with the equation \(t=5 n-3,\) where \(t\) is the number of toothpicks in Stage \(n .\) a. Explain how each part of the equation is related to the toothpick pattern. b. How many toothpicks would Evan need for Stage 10\(?\) For Stage 100\(?\) c. Evan used 122 toothpicks to make one stage of his pattern. Write and solve an equation to find the stage number. d. Is any stage of the pattern composed of 137 toothpicks? Why or why not? e. Is any stage of the pattern composed of 163 toothpicks? Why or why not? f. Evan has 250 toothpicks and wants to make the largest stage of the pattern he can. What is the largest stage he can make? Explain your answer.

Solve each equation or inequality. a. \(1-x=0\) b. \(1-x<0\) c. \(1-x>0\)

Wednesday nights are special at the video arcade: customers pay \(\$ 3.50\) to enter the arcade and then only \(\$ 0.25\) to play each game. Roberto brought \(\$ 7.50\) to the arcade and still had some money when he left. Write an inequality for this situation, using \(n\) to represent the number of games Roberto played.

When you draw a graph, you have to decide the range of values to show on each axis. Each exercise below gives an equation and a range of values for the \(x\) -axis. Use an inequality to describe the range of values you would show on the \(y\) -axis, and explain how you decided. (It may help to try drawing the graphs.) $$y=x^{2}+1 \text { when }-5 \leq x \leq 5$$

Write each equation in the form \(y=m x+b.\) $$-2 y=14 x+\frac{1}{2}(6 x+12)$$
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