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Chapter 1: Problem 26

Find the value of each exponential expression. $$ (0.5)^{4} $$

Short Answer

Expert verified
0.0625

Step by step solution

01

Understand the Base and Exponent

Identify the base and the exponent in the expression. Here, the base is 0.5 and the exponent is 4.
02

Write the Exponential Expression as a Product

The expression \( (0.5)^4 \) means multiplying 0.5 by itself 4 times: \( (0.5) \times (0.5) \times (0.5) \times (0.5) \).
03

Calculate the Product

Multiply the numbers step-by-step: \[ (0.5) \times (0.5) = 0.25 \] \[ 0.25 \times (0.5) = 0.125 \] \[ 0.125 \times (0.5) = 0.0625 \] So, \( (0.5)^4 = 0.0625 \).

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Key Concepts

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

Understanding Base and Exponent
In an exponential expression, you will encounter two key components: the base and the exponent. The base is the number that you multiply by itself. The exponent tells you how many times to multiply the base.
For example, in the expression \((0.5)^4\), the base is 0.5. The exponent is 4, so we will multiply 0.5 by itself 4 times.
If the base were different, such as 2 in \(2^3\), then the base 2 would be multiplied by itself 3 times. It's essential to identify and understand these parts to work successfully with exponential expressions.
The Product of Powers
The phrase 'product of powers' refers to the result you get when you multiply a base raised to an exponent multiple times. This can seem complex at first, but breaking it into steps makes it simpler.
Take the expression \((0.5)^4\). This means you need to find the product of 0.5 multiplied by itself four times. Let's list it down:
  • \((0.5) \times (0.5) = 0.25\)
  • \(0.25 \times (0.5) = 0.125\)
  • \(0.125 \times (0.5) = 0.0625\)
By following this approach, you can easily find the product of powers. Always ensure to multiply step-by-step to avoid mistakes.
Step-by-Step Multiplication
To solve an exponential expression, breaking the process into smaller, manageable steps is crucial. For example, in \((0.5)^4\), we follow these steps:
  • First, recognize the base (0.5) and exponent (4).
  • Next, write it as repeated multiplication: \((0.5) \times (0.5) \times (0.5) \times (0.5)\).
  • Multiply two 0.5s together: \(0.5 \times 0.5 = 0.25\).
  • Multiply the result by 0.5: \(0.25 \times 0.5 = 0.125\).
  • Multiply this result by 0.5 again: \(0.125 \times 0.5 = 0.0625\).
By following these steps, we can confirm that \((0.5)^4 = 0.0625\). Taking it one step at a time prevents errors and ensures accuracy in your calculations.

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Most popular questions from this chapter

A manufacturer has fixed costs of \(\$ 1000\) to produce widgets. Each widget costs \(\$ 5\) to make. The fixed cost to produce gadgets is \(\$ 750,\) and each gadget costs \(\$ 3\) to make. Write an expression for the cost to make \(x\) widgets. (Hint: The cost will be the sum of the fixed cost and the cost per item times the number of items.)

Write a numerical expression for each phrase and simplify. \(-2\) added to the sum of \(-18\) and 11

To find the average (mean) of a group of numbers, we add the numbers and then divide the sum by the number of terms added. For example, to find the average of \(14,8,3,9,\) and \(1,\) we add them and then divide by 5. $$ \frac{14+8+3+9+1}{5}=\frac{35}{5}=7 \leftarrow \text { Average } $$ Find the average of each group of numbers. \(23,18,13,-4,\) and \(-8\)

Perform each indicated operation. $$ -7.6247-[(-3.9928+1.42773)-(-2.80981)] $$

Find each difference. $$ -6.4-3.5 $$
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