/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 26 Evaluate each expression, or sta... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 26

Evaluate each expression, or state that the expression is not a real number. $$\sqrt{16-25}$$

Short Answer

Expert verified
The expression is not a real number.

Step by step solution

01

Calculate inside the square root

To begin, the value inside the square root needs to be calculated: \(16-25\). The result of this calculation is -9.
02

Evaluate the square root

The next step is to take the square root of -9. However, as one may recall, within the set of real numbers, there are no square roots of negative numbers. Hence, this expression is not a real number.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Negative Numbers
Negative numbers are numbers that are less than zero. They are represented with a minus sign (-) in front. For example, -1, -9, and -25 are negative numbers. Negative numbers are an essential part of mathematics and can be found in various situations.
  • They appear in everyday life, like temperatures below zero or owing money.
  • In mathematics, they can affect different operations like addition, subtraction, and, as seen in this exercise, impacts the outcome of square roots.
When dealing with square roots, negative numbers present a particular challenge. The reason is that within the system of real numbers, you cannot find a number which when squared gives a negative result. In simpler terms, there is no real number you can multiply by itself to get a negative number. This is why the square root of -9 does not exist within real numbers.
Real Numbers
Real numbers are any numbers that can be found on the number line. This includes all the positive integers, negative numbers, fractions, and decimals. It's an extensive set that encompasses:
  • Natural numbers: 0, 1, 2, 3, and so forth.
  • Integers: ..., -3, -2, -1, 0, 1, 2, 3, ...
  • Rational numbers: Numbers that can be expressed as fractions, like -2/3 or 0.75.
  • Irrational numbers: Numbers that cannot be expressed as a simple fraction, like \(\pi\) or \(\sqrt{2}\).
While real numbers are quite comprehensive, they do not include imaginary numbers. Imaginary numbers are used to express the square root of a negative number, something that cannot be calculated within the real numbers. For this reason, when faced with the square root of a negative number in the context of real numbers, the answer is that it is not a real number.
Mathematical Expressions
Mathematical expressions are a combination of numbers, operators (like +, -, 脳, 梅), variables, and sometimes exponents or roots. They represent a value or a relationship in mathematics. Evaluating expressions often involves following a set of steps or rules, such as the order of operations.
  • In our exercise, the expression \(\sqrt{16-25}\) must first be simplified inside the root.
  • The operation inside the root (\(16-25\)) results in the calculation of -9 before taking the square root.
  • Due to the presence of a negative result inside the square root, a decision must be made about the nature of the solution.
Expressions are crucial in mathematics as they provide a concise way to represent mathematical ideas and solve problems. They require understanding the underlying principles and the implications of operations involved, such as recognizing when an expression consists of operations that lead to results that are not part of the real numbers set.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

In Exercises \(53-74\), rationalize each denominator. Simplify, if possible. $$\frac{6}{\sqrt{6}+\sqrt{3}}$$

The territorial area of an animal in the wild is the area of the region to which the animal confines its movements. The formula $$T=W^{1.41}=W^{\frac{141}{100}}=\sqrt[190]{W^{141}}$$ models the territorial area, \(T\), in square miles, in terms of an animal's body weight, \(W\), in pounds. a. Use a calculator to fill in the table of values, rounding \(T\) to the nearest whole square mile. $$\begin{array}{l|l|l|l|l|l|l|l}\hline \boldsymbol{W} & \mathbf{0} & \mathbf{2 5} & \mathbf{5 0} & \mathbf{1 5 0} & \mathbf{2 0 0} & \mathbf{2 5 0} & \mathbf{3 0 0} \\\\\hline \boldsymbol{T}=\boldsymbol{W}^{1.41} & & & & & & & \\\\\hline\end{array}$$ b. Use the table of values to graph \(T=W^{1.41} .\) What does the shape of the graph indicate about the relationship between body weight and territorial area? c. Verify your hand-drawn graph by using a graphing utility to graph the model.

Make Sense? In Exercises \(90-93,\) determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. Without using a calculator and knowing that \(\sqrt{2} \approx 1.4142\) rationalizing the denominator of \(\frac{1}{\sqrt{2}}\) makes division to obtain a decimal approximation for \(\frac{1}{\sqrt{2}}\) easier to perform.

Solve: \(\quad 6 x^{2}-11 x+5=0 .\) (Section 6.6, Example 2)

Simplify by first writing the expression in radical form. If applicable, use a calculator to verify your answer. $$4^{\frac{5}{2}}-8^{\frac{2}{3}}$$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.