/*! 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 106 Find the value of the expression... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 106

Find the value of the expression when \(x_{1}=2, x_{2}=4, y_{1}=-3\) \(y_{2}=2\) $$ \sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}} $$

Short Answer

Expert verified
The value is \(\sqrt{29}\).

Step by step solution

01

Understand the Expression

The given expression \( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \) represents the Euclidean distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\). We need to substitute the given values into this formula to find the answer.
02

Substitute Values for x1 and x2

Substitute \(x_1 = 2\) and \(x_2 = 4\) into the equation: \((x_2 - x_1)^2 = (4 - 2)^2 = 2^2 = 4\).
03

Substitute Values for y1 and y2

Substitute \(y_1 = -3\) and \(y_2 = 2\) into the equation: \((y_2 - y_1)^2 = (2 - (-3))^2 = (2 + 3)^2 = 5^2 = 25\).
04

Compute the Sum of Squares

Add the results from Step 2 and Step 3: \(4 + 25 = 29\).
05

Take the Square Root

Calculate the square root of the sum obtained in Step 4: \(\sqrt{29}\).
06

Solutions and Conclusion

The value of the expression is \(\sqrt{29}\), which is the exact distance between the points. This is an irrational number that cannot be simplified further.

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.

Square Root
When dealing with mathematical expressions, the square root plays a crucial role in many calculations. The square root of a number is a value, that when multiplied by itself, will give the original number. In symbols, if you see \( \sqrt{x} \), it means finding a number \( y \) such that \( y \times y = x \).
  • The operation of finding the square root is the inverse of squaring a number.
  • It is particularly important in geometry and algebra, especially when calculating distances.
In the context of our exercise, the square root is used to determine the Euclidean distance. After computing the sum of squares, the square root turns this sum into an understandable measurement of distance.
Substitution
Substitution is a simple yet powerful tool in mathematics. It involves replacing variables with specific values. This technique helps in evaluating expressions accurately.
  • Locate the variable in the expression.
  • Replace the variable with the given number.
  • Perform the arithmetic operations as required.
In the exercise, we substitute two sets of variables, \( x_1, x_2 \) and \( y_1, y_2 \), with their respective numbers. This transforms the general formula into a specific equation that can easily be solved.
Irrational Numbers
Irrational numbers are numbers that cannot be expressed as a simple fraction. They are real numbers with non-repeating, non-ending decimals.
  • Common examples include \( \pi \) and \( \sqrt{2} \).
  • These numbers often arise when taking the square root of non-perfect squares.
In our exercise, \( \sqrt{29} \) is the result, a number that cannot be expressed exactly as a fraction, making it irrational. Understanding irrational numbers helps in recognizing when an exact decimal representation is not possible.
Distance Formula
The distance formula is essential in coordinate geometry for finding the distance between two points on a plane. It is derived from the Pythagorean Theorem. The formula is written as \( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).
  • The difference \( x_2 - x_1 \) measures horizontal distance.
  • The difference \( y_2 - y_1 \) measures vertical distance.
  • The formula then combines these into the actual straight-line distance.
Using this formula with specific values offers a precise measurement of how far apart the two points are. It's widely useful in various applications like navigation, physics, and engineering.

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

Simplify each expression. See Examples I through 11 . $$ 2\\{-1+3[7-4(-10+12)]\\} $$

Use a calculator to approximate each square root. simplify the expression. Round answers to four decimal places. \(\sqrt{7.9}\)

Simplify each expression. $$ 6.5 y-4.4(1.8 x-3.3)+10.95 $$

Simplify each expression. See Examples \(11,14,\) and \(15 .\) $$ 2(3 x+7) $$

Simplify each expression. See Examples \(11,14,\) and \(15 .\) $$ 4\left(6 n^{2}-3\right)-3\left(8 n^{2}+4\right) $$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

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.