/*! 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 41 \(39-48=\) Simplify the expressi... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 41

\(39-48=\) Simplify the expression. $$ \sqrt{125}-\sqrt{45} $$

Short Answer

Expert verified
The simplified expression is \(2\sqrt{5}\).

Step by step solution

01

Simplify the Radical Expressions

Start by simplifying each radical expression individually. First, look at \(\sqrt{125}\). We know that \(125 = 25 \times 5\), and can thus rewrite it as \(\sqrt{25 \times 5}\). Because \(25\) is a perfect square, we can simplify further: \(\sqrt{25 \times 5} = \sqrt{25} \times \sqrt{5} = 5\sqrt{5}\).
02

Simplify the Second Radical

Next, simplify \(\sqrt{45}\). Notice that \(45 = 9 \times 5\), so we can rewrite it as \(\sqrt{9 \times 5}\). Since \(9\) is a perfect square, we further simplify it to \(\sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3\sqrt{5}\).
03

Subtract the Simplified Terms

Now, with both radicals simplified, we have \(5\sqrt{5} - 3\sqrt{5}\). Since both terms have \(\sqrt{5}\) in common, we can combine them: \(5\sqrt{5} - 3\sqrt{5} = (5 - 3)\sqrt{5} = 2\sqrt{5}\).

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.

Radical Expressions
Radical expressions involve the use of roots, commonly square roots. Simplifying them means rewriting them in their simplest form. Consider the expression \( \sqrt{125} - \sqrt{45} \). We break each part into its prime factors or perfect squares to simplify:
  • For \( \sqrt{125} \), recognize that 125 can be factored as \( 25 \times 5 \).
  • Since 25 is a perfect square, we simplify \( \sqrt{25} \) to 5, resulting in \( 5\sqrt{5} \).
  • Similarly, for \( \sqrt{45} \), use the factors \( 9 \times 5 \).
  • The perfect square 9 simplifies to 3, giving us \( 3\sqrt{5} \).
Each step involves breaking down the number under the radical sign into its simplest parts, making it easier to perform operations on radical expressions later on.
Perfect Squares
A perfect square is an integer that is the square of another integer. Recognizing these can greatly help in simplifying radical expressions. In our exercise, we identified:
  • 25 as a perfect square because \( 25 = 5^2 \).
  • 9 as a perfect square because \( 9 = 3^2 \).
These perfect squares allow for simplification under the radical sign when simplifying expressions. By factoring numbers into elements where at least one is a perfect square, you can take its root out of the radical. This process dramatically simplifies complex expressions and is a crucial skill for ease and accuracy when working through algebraic problems.
Combining Like Terms
When dealing with radical expressions, combining like terms is similar to how you handle variables in algebra. After simplifying radicals, you may find terms with the same radical part - these are like terms. In our case:
  • We have \( 5\sqrt{5} \) and \( 3\sqrt{5} \).
  • Both terms have \( \sqrt{5} \), which can be thought of like a variable \( x \).
  • Therefore, you can combine them: \( 5\sqrt{5} - 3\sqrt{5} = (5-3)\sqrt{5} = 2\sqrt{5} \).
The key is to ensure the radical part of each term is identical. Once confirmed, treat the radical as if it were a common variable and combine the coefficients. This technique helps in simplifying expressions to their most condensed form, allowing for more intuitive and streamlined computation.

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 the rational expression. $$ \frac{x^{2}-x-2}{x^{2}-1} $$

Perform the multiplication or division and simplify. $$ \frac{x^{2}-x-6}{x^{2}+2 x} \cdot \frac{x^{3}+x^{2}}{x^{2}-2 x-3} $$

Perform the multiplication or division and simplify. $$ \frac{x^{2}+2 x-3}{x^{2}-2 x-3} \cdot \frac{3-x}{3+x} $$

Simplify the rational expression. $$ \frac{4\left(x^{2}-1\right)}{12(x+2)(x-1)} $$

Perform the multiplication or division and simplify. $$ \frac{t-3}{t^{2}+9} \cdot \frac{t+3}{t^{2}-9} $$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Pure 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.