/*! 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 58 In Exercises \(55-68,\) multiply... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 58

In Exercises \(55-68,\) multiply and, if possible, simplify. $$\sqrt{8 x} \cdot \sqrt{10 y}$$

Short Answer

Expert verified
The simplified result of the given expressions is \(4 \sqrt{5xy}\)

Step by step solution

01

Combine the Expressions

First, let's use the multiplication property of square roots, which says that \(\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}\). Therefore, \(\sqrt{8x} \cdot \sqrt{10y} = \sqrt{8x * 10y}\)
02

Simplify the Expression

Next, simplify the expression under the square root: \(\sqrt{8x * 10y} = \sqrt{80xy}\)
03

Factor Out Perfect Squares

Factor out perfect squares from the expression within the square root. This allows us to find \(\sqrt{80xy} = \sqrt{16*5xy} = \sqrt{16}*\sqrt{5xy} = 4\sqrt{5xy}\)

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 Roots
A square root is a special value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because 3 multiplied by 3 equals 9. The symbol for square root is \(\sqrt{}\). It鈥檚 like asking, "What number squared gives us this number?"
  • Origin of Square Roots: The concept dates back to ancient civilizations where solving equations and understanding shapes required this operation.
  • Properties of Square Roots: One of the most useful properties is the multiplication property. It allows us to break down or combine roots: \(\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}\).
For example, in the given exercise, we use this property to combine \(\sqrt{8x}\) and \(\sqrt{10y}\) into \(\sqrt{80xy}\). This technique simplifies the mathematical process when working with roots.
Simplifying Expressions
When you simplify expressions, your goal is to make them easier to work with, while keeping their value the same. Simplifying can involve merging like terms, factoring, and using mathematical properties like the distribution of multiplication over addition.
  • Benefits of Simplification: By simplifying, mathematicians and students can more easily interpret and solve expressions, exposing underlying patterns or simpler forms.
  • Steps to Simplify: Commonly, you'll combine terms, factor numbers, or break them down into prime factors. This then allows for the further extraction of square roots.
In the problem, we first combine \(8x\) and \(10y\) under one radical to form \(\sqrt{80xy}\), then extract any perfect squares to simplify further. Simplification not only provides a clearer answer but also makes further calculations more straightforward.
Perfect Squares
A perfect square is a number that is the square of an integer. In simpler terms, it鈥檚 the product of a number multiplied by itself. For example, \(16\) is a perfect square because \(4 \times 4 = 16\).
  • Identifying Perfect Squares: Recognizing these in expressions makes simplifying much easier since square roots of perfect squares are whole numbers.
  • Using Perfect Squares to Simplify: When working with radicals, factor the number to find any perfect squares. For instance, you can split \(80\) into \(16 \times 5\); here, \(16\) is the perfect square.
In the exercise, we identify \(16\) as a perfect square within \(80\). By expressing \(\sqrt{80xy}\) as \(\sqrt{16} \cdot \sqrt{5xy}\), we can simplify this to \(4\sqrt{5xy}\). This step emphasizes the power of recognizing and utilizing perfect squares in mathematical simplification.

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 by first writing the expression in radical form. If applicable, use a calculator to verify your answer. $$27^{\frac{2}{3}}+16^{\frac{3}{4}}$$

Simplify by first writing the expression in radical form. If applicable, use a calculator to verify your answer. $$\left(\frac{4}{25}\right)^{-\frac{1}{2}}$$

In Exercises \(94-97,\) determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\frac{3 \sqrt{x}}{x \sqrt{6}}=\frac{\sqrt{6 x}}{2 x} \text { for } x>0$$

Fill in the box to make the statement true: \(\frac{4}{2+\sqrt{3}}=8-4 \sqrt{3}\)

Simplify each expression. Write answers in exponential form with positive exponents only. Assume that all variables represent positive real numbers. $$\left(\frac{x^{\frac{2}{5}}}{x^{\frac{6}{5}} \cdot x^{\frac{3}{5}}}\right)^{5}$$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

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.