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Chapter 3: Problem 88

Evaluate each expression. $$ 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 $$

Short Answer

Expert verified
1024

Step by step solution

01

Identify the Expression

The expression to evaluate is 4 multiplied by itself five times: ewline\[ 4 \times 4 \times 4 \times 4 \times 4 \]
02

Express in Exponential Form

Rewrite the expression using exponential notation. Since 4 is multiplied by itself five times, it can be written as: \ \[ 4^5 \]
03

Calculate the Expression

Now, calculate the value of \ \[ 4^5. \] To do this, multiply 4 by itself five times: \[ 4^5 = 4 \times 4 \times 4 \times 4 \times 4 = 4 \times 4 = 16 \] \[ 16 \times 4 = 64 \] \[ 64 \times 4 = 256 \] \[ 256 \times 4 = 1024 \] So, \ \[ 4^5 = 1024 \]

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Key Concepts

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

evaluate expressions
Evaluating expressions means simplifying them to find their value. It's like solving a puzzle where the goal is to reach the simplest form or numerical value. In our example, we started with an expression involving multiple multiplications of the same number: 4 multiplied by itself five times.

To evaluate such expressions, you can follow these steps:
  • Identify the expression you need to evaluate.
  • If possible, rewrite it in a simpler form using mathematical notations like exponents.
  • Carry out the required operations step-by-step to find the final value.
By breaking down the problem into smaller steps, evaluating becomes easier and more manageable.
exponents
An exponent is a compact way to represent repeated multiplication of the same number. When you see an expression like \(4^5\), it means 4 multiplied by itself five times. Here are some key concepts:
  • Base: The number being multiplied (in this case, 4).
  • Exponent: The number of times the base is multiplied by itself (in this case, 5).
So, \(4^5\) means \(4 \times 4 \times 4 \times 4 \times 4\).

Using exponents can simplify complex multiplications and make calculations more straightforward. It's a powerful notation that appears frequently in mathematics, so understanding it well is crucial.
multiplication
Multiplication is one of the basic arithmetic operations. It's a method of adding a number to itself a specified number of times. For example, \(4 \times 4\) means adding 4 to itself once, which gives 16.

In our example, we performed multiplication of 4 several times:
  • First, we calculated \(4 \times 4 = 16\).
  • Next, \(16 \times 4 = 64\).
  • Then, \(64 \times 4 = 256\).
  • Finally, \(256 \times 4 = 1024\).
Understanding how to multiply numbers step-by-step is essential, especially when dealing with large numbers or expressions involving exponents.

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Most popular questions from this chapter

Concept Check In Exercises \(55-62,\) describe what the graph of each linear equation will look like in the coordinate plane. (Hint: Rewrite the equation if necessary so that it is in a more recognizable form.) $$ 2 x=4 y $$

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It costs a flat fee of 20 dollars plus 5 dollars per day to rent a pressure washer. Therefore, the cost to rent the pressure washer for \(x\) days is given by $$ y=5 x+20 $$ where \(y\) is in dollars. Express each of the following as an ordered pair. (a) When the washer is rented for 5 days, the cost is 45 dollars. (b) I paid 50 dollars when I returned the washer, so I must have rented it for 6 days.

The table shows the rate (in percent) at which 2-year college students (public) completed a degree within 3 years. $$ \begin{array}{|c|c|} \hline \text { Year } & {\text { Percent }} \\ \hline 2002 & {31.6} \\ {2003} & {30.1} \\ {2004} & {29.0} \\ {2005} & {27.5} \\ {2006} & {26.6} \\ {2007} & {26.9} \\ \hline \end{array} $$ (a) Write the data from the table as ordered pairs \((x, y),\) where \(x\) represents the year and \(y\) represents the percent. (b) What does the ordered pair \((2007,26.9)\) mean in the context of this problem? (c) Make a scatter diagram of the data, using the ordered pairs from part (a) and the given grid. (d) Describe the pattern indicated by the points on the scatter diagram. What is happening to rates at which 2 -year college students complete a degree within 3 years?
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