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Chapter 1: Problem 66

Evaluate each expression. (a) \(|\sqrt{5}-5|\) (b) \(|10-\pi|\)

Short Answer

Expert verified
(a) 2.764, (b) 6.85841

Step by step solution

01

Evaluate the inside of the absolute value for Part (a)

Calculate the expression inside the absolute value.\[|\sqrt{5} - 5|\]Firstly, find the approximate value of \(\sqrt{5}\). Since \(\sqrt{5} \approx 2.236\), substitute this value:\[2.236 - 5 = -2.764\]
02

Apply absolute value for Part (a)

Find the absolute value of the result from Step 1.\[|-2.764| = 2.764\]Therefore, the evaluated expression for Part (a) is 2.764.
03

Evaluate the inside of the absolute value for Part (b)

Calculate the expression inside the absolute value.\[|10 - \pi|\]Since \(\pi \approx 3.14159\), substitute this value:\[10 - 3.14159 = 6.85841\]
04

Apply absolute value for Part (b)

Find the absolute value of the result from Step 3.\[|6.85841| = 6.85841\]Therefore, the evaluated expression for Part (b) is 6.85841.

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Key Concepts

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

Evaluating Expressions
Evaluating expressions involves performing operations inside a mathematical expression to arrive at a conclusion or answer. In the context of this exercise, you start by solving the part of the expression inside the absolute value brackets. For instance, consider the expression \( \sqrt{5} - 5 \). You first find the square root of 5, then subtract 5 from that result, and lastly, apply the absolute value.
  • Identify the operation: Look at what needs to be done first. Here, it's finding the square root.
  • Calculate: Do the subtraction after calculating the square root or pi value.
  • Apply absolute value: Change any negative result into its positive equivalent.
To evaluate effectively, follow these steps carefully for clarity and accuracy.
Square Roots
A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 25 is 5 because \( 5 \times 5 = 25 \). In this exercise, we specifically look at the square root of 5.
  • Approximation: Since \( \sqrt{5} \) is not a whole number, we find it approximately as 2.236. This involves understanding that root values often require approximation to decimals for easier calculations.
  • Justification: Even if the result turns out to be a decimal, it's important to cross-check by estimating which two perfect squares the number lies between (for \( \sqrt{5} \), those are 4 and 9).
Understanding square roots helps in solving expressions more accurately, especially when they form part of larger equations.
Approximation of Pi
Pi (often represented by the symbol \( \pi \)) is a mathematical constant that represents the ratio of a circle's circumference to its diameter. While the true value of \( \pi \) is infinite and non-repeating, for calculations, we use approximations.
  • Common Approximations: \( \pi \) is usually approximated as 3.14 or 3.14159 for more precision. This helps in making computations simpler.
  • Usage of Pi: In Part (b) of the exercise, pi is used to subtract from 10. This illustrates how pi often comes into calculations involving geometry or when precision is important.
Approximations simplify complex numbers into forms that are useful for manual calculations, keeping the results both practical and manageable.

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Most popular questions from this chapter

Factor the expression completely. $$12 x^{3}+18 x$$

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