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Chapter 7: Problem 87

Use a calculator to approximate each radical to three decimal places. $$ \sqrt[5]{23.8} $$

Short Answer

Expert verified
The approximate value of \(\root{5}{23.8}\) is 1.742.

Step by step solution

01

- Understand the problem

Identify the radical expression that needs to be approximated. In this case, it is \(\root{5}{23.8}\).
02

- Use a calculator

Input the expression into a scientific calculator. Most calculators have a function for computing roots. Look for the \(\root{x}{y}\) or \(\root[n]{x}\) button, sometimes represented as \(y^{1/n}\).
03

- Input the values

Enter the value 23.8 and set the root to 5. This might be done using the format \(23.8^{(1/5)}\) depending on the calculator.
04

- Compute and round

Calculate the value and round the result to three decimal places. The calculator shows an output of approximately 1.742.

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Key Concepts

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

fifth root calculation
Calculating the fifth root of a number might sound tricky, but with a clear understanding, it becomes a straightforward task. When we talk about the fifth root of a number, we're looking for the value that, when multiplied by itself five times, equals the original number. For example, if you have \(\root{5}{243} = 3\), it means that 3 multiplied by itself five times equals 243: \(3 \times 3 \times 3 \times 3 \times 3 = 243\). To find the fifth root of 23.8, \(\root{5}{23.8}\), we'll use a calculator for a precise approximation.
rounding decimal places
Once you have calculated the fifth root using the calculator, you will usually see a long decimal number. Rounding is a crucial step to simplify this result. Rounding to three decimal places means you look at the fourth decimal place to decide. If it is 5 or higher, round up the third decimal place by 1. If it is lower than 5, keep the third decimal place as is. For instance, if the calculator shows 1.74192, you check the fourth decimal place (which is 9 in this case) and round the number to 1.742. This simplified number is much easier to use and understand.
scientific calculator usage
Scientific calculators are essential for computing more complex mathematical expressions, such as radical expressions. To find the fifth root of 23.8 using a scientific calculator, you will generally follow these steps:
  • Locate the root function, which might be represented as \(\root{x}{y}\) or \({y^{1/n}}\).
  • Enter the base number (23.8) followed by the operation for the fifth root (represented as 1/5).
  • Calculate to get the result.
Different calculators might have slightly different methods, so consult your calculator's manual if you have any trouble.
radical approximation
Approximating radicals involves finding a decimal value close to the actual radical value. This is often necessary as many radicals do not result in a whole number and can be irrational numbers. When you approximate \(\root{5}{23.8}\), you use a calculator for more precision. After computing, you often get a long decimal, like 1.74192. You then approximate by rounding it to three decimal places, giving you 1.742. This approximation helps in simplifying and making the radical easier to work with in further calculations or practical applications.

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Most popular questions from this chapter

Simplify. Assume that all variables represent positive real numbers. \(-\sqrt[3]{27 t^{12}}\)

Write each radical as an exponential and simplify. Leave answers in exponential form. Assume that all variables represent positive numbers. $$ \sqrt{\sqrt[3]{\sqrt[4]{x}}} $$

Rationalize each denominator. Assume that all variables represent positive real numbers and that no denominators are 0. $$ \frac{3 \sqrt{x}}{\sqrt{x}-2 \sqrt{y}} $$

Find the equation of a circle satisfying the given conditions. Center: (-8,-5)\(;\) radius: \(\sqrt{5}\)

Rationalize each denominator. Assume that all variables represent positive real numbers and that no denominators are 0. $$ \frac{5 \sqrt{k}}{2 \sqrt{k}+\sqrt{q}} $$
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