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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

Consider the inequality \(x^{3} \leq 27.\) a. Express the solution of \(x^{3} \leq 27\) as an inequality. b. Graph the solution on a number line.

Rewrite each expression as simply as you can. $$\left(a^{m}\right)^{n} \cdot\left(b^{3}\right)^{0}$$

Recall that for linear equations, first differences are constant; and that for quadratic equations, second differences are constant. Determine whether the relationship in each table could be linear, quadratic, or neither. $$ \begin{array}{|c|c|}\hline x & {y} \\ \hline-3 & {0} \\ {-2} & {-0.5} \\\ {-1} & {-1} \\ {0} & {-1.5} \\ {1} & {-2} \\ {2} & {-2.5} \\\ \hline\end{array} $$

Solve the systems of equations in Exercises \(8-11\) by elimination, and check your solutions. Give the following information: \(\cdot\) which variable you eliminated \(\cdot\) whether you added or subtracted equations \(\bullet\) the solution $$ \begin{array}{l}{9 s+2 t=3} \\ {4 s+2 t=8}\end{array} $$

List six whole numbers that satisfy the inequality \(n-2>5.\)
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