/*! 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 107 Explain the multiplication prope... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 107

Explain the multiplication property of radicals.

Short Answer

Expert verified
The Multiplication Property of Radicals is a principle that states \(\sqrt[a]{m} * \sqrt[a]{n} = \sqrt[a]{m*n}\), where 'm' and 'n' are real numbers and 'a' is a positive integer. It means that the product of two radicals having the same index can be written as a single radical.

Step by step solution

01

Definition

The Multiplication Property of Radicals states that for any real numbers 'm' and 'n' and any positive integer 'a': \(\sqrt[a]{m} * \sqrt[a]{n} = \sqrt[a]{m*n}\). This means that the product of two radicals with the same index can be written as one radical.
02

- Illustration with Examples

Let's look at an example: \(\sqrt{2} * \sqrt{3}\) can be written as \(\sqrt{2*3}\), which simplifies to \(\sqrt{6}\). Similarly, if we have \(\sqrt[3]{2} * \sqrt[3]{4}\), it can be simplified as \(\sqrt[3]{2*4}\), which simplifies to \(\sqrt[3]{8}\). So, both examples follow the Multiplication Property of Radicals.
03

- Important Note

An important note to remember is that this property holds only when the indices of the radicals being multiplied are the same. Also, the radicands (the values within the radical symbol) should be positive, or the result could lead to imaginary or undefined numbers.

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.

Algebra
Algebra is a fundamental aspect of mathematics. It allows us to describe mathematical relationships and work with unknown values. With algebra, we use symbols or letters to represent numbers or quantities in equations and formulas. This makes it easier to solve problems and understand mathematical concepts.

When dealing with radicals, algebra allows us to manipulate and simplify expressions using properties like the multiplication property of radicals. By understanding algebraic principles, we can simplify complex expressions and solve equations that involve radicals.
  • Symbols commonly used in algebra include letters like 'x', 'y', and 'z'
  • Equations often involve operations like addition, subtraction, multiplication, and division
  • Algebra helps in finding the values of unknowns in equations
Radical Expressions
Radical expressions are mathematical expressions that include a radical symbol, \(\sqrt{}\), which represents the root of a number. The most common radical expression is the square root, but radicals can have different indices, such as cube roots, fourth roots, and so on.

In algebra, radical expressions are important because they allow us to work with equations and functions involving roots. To simplify or solve these expressions, properties like the multiplication property of radicals are useful. For example, \(\sqrt[3]{27}\) simplifies to 3 because 3 is the number that, when multiplied by itself twice, gives 27.
  • Radicals can simplify complex problems, especially when dealing with square roots
  • The index tells us which root the radical represents, like square (2) or cube (3)
  • Simplifying radical expressions often involves removing perfect squares or cubes
Index of a Radical
The index of a radical is a small number written just outside and slightly above the radical sign. It indicates which root of the number you are taking. For instance, in \(\sqrt[3]{x}\), the 3 is the index, showing it's a cube root.

Understanding the index of a radical is essential for applying the multiplication property of radicals. This property only works smoothly when radicals share the same index. If two different indices are involved, further steps would be necessary to combine or simplify the expressions.
  • The default index is 2, which means square root
  • Index determines the root you will be calculating, like cube (3) or fourth (4)
  • Consistency in index is crucial for simplifying expressions correctly
Real Numbers
Real numbers include every number you can think of, including whole numbers, fractions, and decimals. They also incorporate both positive and negative numbers as well as zero. Essentially, real numbers encompass all possible numbers along the number line.

In the context of the multiplication property of radicals, ensuring we're working with real numbers is crucial. This is because, with negative numbers under even roots (like square roots), the outcome can be imaginary or undefined.
  • Real numbers range from negative infinity to positive infinity
  • They exclude imaginary numbers, such as the square root of negative 1
  • Understanding real numbers helps ensure you're working in a valid number system for your calculations

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 radical expression. If the answer is not exact, round to the nearest hundredth. All variables represent positive values $$\sqrt[5]{\frac{1}{32}}$$

Graph each function. $$f(x)=1+\sqrt[3]{x}$$

Simplify. $$\sqrt{72}+\sqrt{200}$$

Multiply. In square roots, all variables represent positive values. SEE EXAMPLE 2 . (OBJECTIVE 1) $$\sqrt[3]{5}(\sqrt[3]{25}+3)$$

Use scientific notation to simplify \(\sqrt{0.00000004}\).
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Calculus

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.