/*! 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 8 Multiply. $$ \sqrt{10} \sqrt... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 8

Multiply. $$ \sqrt{10} \sqrt{3} $$

Short Answer

Expert verified
The result is \( \sqrt{30} \).

Step by step solution

01

Identify the Multiplicands

Recognize that we need to multiply two square roots: \ \ \( \sqrt{10} \) and \( \sqrt{3} \).
02

Use the Property of Square Roots

Apply the property of square roots which states that \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \). Here, \( a = 10 \) and \( b = 3 \).\ \ Therefore, \( \sqrt{10} \cdot \sqrt{3} = \sqrt{10 \cdot 3} \).
03

Perform the Multiplication Inside the Square Root

Calculate the product inside the square root: \ \ \( 10 \cdot 3 = 30 \).\ This simplifies the expression to: \ \( \sqrt{30} \).

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.

Properties of Square Roots
Square roots have several properties that make them easier to work with, especially when multiplying or dividing them. One of the most useful properties is the product rule, which states:
\[\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\]
This means that you can multiply the numbers inside the square roots first, and then take the square root of the resulting product. For example, when multiplying \(\sqrt{10}\) and \(\sqrt{3}\), you can combine them inside one square root:
\( \sqrt{10} \cdot \sqrt{3} = \sqrt{10 \cdot 3} = \sqrt{30} \).
This property simplifies calculations and is very helpful in algebra and dealing with radical expressions.
Radical Expressions
Radical expressions involve roots, such as square roots. A square root of a number is a value that, when multiplied by itself, gives the original number. For example, \( \sqrt{9} = 3 \) because \( 3^2 = 9 \).
Radical expressions can include numbers, variables, or both under the square root symbol. When dealing with radical expressions, it's important to know how to simplify and manipulate them using different properties, including the multiplication property we used earlier. This helps to handle more complex algebraic expressions efficiently.
Simplifying Radicals
Simplifying radicals means making the expression under the square root as simple as possible. Often, this involves breaking numbers down into their prime factors. For instance, to simplify \(\sqrt{30}\), you would break down 30 into its prime factors: \( 30 = 2 \cdot 3 \cdot 5 \).
If a number is a perfect square (like 4, 9, 16), its square root is an integer. However, in many cases, such as \( \sqrt{30} \), it is already in its simplest form because 30 does not have any perfect square factors. Simplifying radicals helps in keeping expressions clear and concise, which is particularly useful for solving equations and other algebraic tasks.

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

Outline a procedure that uses the distance formula to determine whether three points, \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right),\) and \(\left(x_{3}, y_{3}\right),\) are collinear (lie on the same line).

Review simplifying rational expressions (Sections\(7.1-7.5)\) Perform the indicated operation and, if possible, simplify. $$ \frac{a^{-1}+b^{-1}}{a b}[7.5] $$

Find a simplified form for \(f(x) .\) Assumex \(\geq 0\). $$f(x)=\sqrt[4]{x^{5}-x^{4}}+3 \sqrt[4]{x^{9}-x^{8}}$$

Ramon incorrectly writes $$\sqrt[5]{x^{2}} \cdot \sqrt{x^{3}}=x^{2 / 5} \cdot x^{3 / 2}=\sqrt[5]{x^{3}}$$ What mistake do you suspect he is making?

Roofing. Kit's home, which is 24 ft wide and 32 ft long, needs a new roof. By counting clapboards that are 4 in. apart, Kit determines that the peak of the roof is 6 ft higher than the sides. If one packet of shingles covers \(100 \mathrm{ft}^{2},\) how many packets will the job require?
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

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.