/*! 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 25 Simplify. $$\sqrt{125}+\sqrt{2... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 25

Simplify. $$\sqrt{125}+\sqrt{245}$$

Short Answer

Expert verified
The simplified form of the expression is \(12\sqrt{5}\)

Step by step solution

01

Factor the numbers under the square root

Express the numbers \(125\) and \(245\) as a product of prime numbers. So, we have \(125 = 5^3\) and \(245 = 5*7^2\)
02

Apply the rule of square roots

Use the rule that \(\sqrt{a^2*b} = a\sqrt{b}\) for each quantity. Thus the expression becomes \(\sqrt{5^3}+\sqrt{5*7^2} = 5\sqrt{5} + 7\sqrt{5}\)
03

Combine like terms

Now combine the two terms. They are like terms because they both include the term \(\sqrt{5}\). Therefore, \(5\sqrt{5} + 7\sqrt{5} = 12\sqrt{5}\)

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.

Prime Factorization
When simplifying square roots, prime factorization is an essential first step. This method involves breaking down a number into its prime number components. Let鈥檚 revisit what prime numbers are. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. The numbers 2, 3, 5, 7, 11 are all examples of prime numbers.

Now, consider how to factor 125 and 245 into their prime components:
  • For 125, begin by dividing it by 5, since 5 is a prime number: 125 梅 5 = 25. Then, factor 25 into 5 脳 5. Hence, 125 = 5 脳 5 脳 5, or 53.
  • For 245, start with the smallest prime number that fits, which is also 5: 245 梅 5 = 49. Notice that 49 is 7 squared, so 245 = 5 脳 7 脳 7, or 5 脳 72.
Through prime factorization, we have expressed these numbers in a form that will simplify the square root calculations.
Square Root Properties
Understanding the properties of square roots can greatly simplify expressions. A key property of square roots is that they can be simplified when radical terms have perfect squares. The property used in this exercise is expressed as: \[\sqrt{a^2 \times b} = a \sqrt{b}.\]This property allows us to "take out" the square root of a perfect square from under the radical.

Let鈥檚 apply this property to our problem:
  • For \(\sqrt{5^3}\), this can be rewritten as \(\sqrt{5^2 \times 5}\). Since \(5^2\) is a perfect square, it simplifies to \(5\sqrt{5}\).
  • For \(\sqrt{5 \times 7^2}\), rewrite as \(\sqrt{7^2 \times 5}\). This simplifies to \(7\sqrt{5}\) because \(7^2\) is a perfect square.
Utilizing square root properties helps in obtaining a simplified expression, moving us closer to our final solution.
Combining Like Terms
Combining like terms is one of the final steps in simplifying expressions. This involves recognizing terms that are similar and adding them together. The term "like terms" refers to terms in an expression that have identical variables and exponents, allowing them to be combined.

In our example, after applying square root properties, we get the terms \(5\sqrt{5}\) and \(7\sqrt{5}\). Notice they both include the radical \(\sqrt{5}\). Since both terms share this radical, they can be combined:
  • Add the coefficients (the numbers in front of the radicals): \(5 + 7 = 12\).
  • The combined term is \(12\sqrt{5}\).
By grouping like terms, we achieve a simplified and more manageable expression suitable for further calculation or interpretation.

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. All variables of square root expressions represent positive numbers. Assume no division by 0. $$\sqrt{\frac{36 m^{2} n^{9}}{100 m n^{3}}}$$

Simplify. All variables in square root problems represent positive values. Assume no division by 0. $$\frac{3}{\sqrt{3}-\sqrt{2}}$$

Simplify each expression. All variables of square root expressions represent positive numbers. Assume no division by 0. $$\frac{1}{5} x^{5} y \sqrt{75 x^{3} y^{2}}$$

Simplify each expression. Assume that all variables represent positive values. Assume no division by 0. SEE EXAMPLE 6. (OBJECTIVE 3) $$\left(32 b^{1 / 2}\right)^{-2 / 5}$$

Simplify each expression. All variables of square root expressions represent positive numbers. Assume no division by 0. $$\sqrt[3]{1,600 x y^{2} z^{3}}$$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Statistics

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.