/*! 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 26 If possible, simplify each radic... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 26

If possible, simplify each radical expression. Assume that all variables represent positive real numbers. $$\sqrt{45}$$

Short Answer

Expert verified
\(\sqrt{45} = 3\sqrt{5}\)

Step by step solution

01

Identify the perfect square factor

The first task is to identify a perfect square factor of 45. The number 45 can be broken down into smaller factors: \[ 45 = 9 imes 5 \] Here, 9 is a perfect square (\(3^2\) = 9).
02

Break down the radical expression

Using the factorization from Step 1, rewrite the original expression by separating the perfect square from the other factor:\[ \sqrt{45} = \sqrt{9 imes 5} \]
03

Apply the square root to the perfect square

Use the property of square roots that says \( \sqrt{a imes b} = \sqrt{a} imes \sqrt{b} \). Here, we take the square root of the perfect square:\[ \sqrt{45} = \sqrt{9} imes \sqrt{5} \]Knowing that \(\sqrt{9} = 3\), you can further simplify it: \[ \sqrt{45} = 3 \times \sqrt{5} \]
04

Simplify the expression if needed

The expression \(3 \times \sqrt{5}\) is already in its simplest form because 5 is a prime number and cannot be simplified under the square root any further.

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.

Perfect Square Factor
A perfect square factor is a number that can be expressed as the square of an integer. These numbers are fundamental when simplifying radical expressions. In our example, we need to simplify \( \sqrt{45} \). The process begins by identifying perfect square factors. We factorize 45 to find its divisors:
  • 45 = 9 脳 5
  • Here, 9 is a perfect square because it equals \(3^2\).
Recognizing perfect square factors simplifies the process of removing them from under the square root. This means you only keep smaller, non-perfect square numbers under the radical sign.
Square Root
The square root function allows us to find a number that, when multiplied by itself, yields the original number. In simple terms, it's the opposite of squaring a number. Every number has two square roots: positive and negative, but we typically use the positive square root when simplifying radicals.
  • The square root of a perfect square like 9 is a whole number: \( \sqrt{9} = 3 \).
  • For non-perfect squares, such as 5, the square root remains in the radical form: \( \sqrt{5} \).
In radical simplification exercises, understanding when a square root can be simplified to a whole number is crucial to making the expression simple.
Radical Expression Simplification
The goal of simplifying a radical expression is to express it in its most reduced form. This often involves factoring out perfect square factors. Here's how we simplify \(\sqrt{45}\) using the principles discussed:
  • Recognize that 45 can be expressed as 9 脳 5.
  • Re-write the expression to separate the perfect square: \(\sqrt{45} = \sqrt{9 \times 5}\).
  • Apply the property of radicals: \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \).
  • Simplify further using \(\sqrt{9} = 3\): \(\sqrt{45} = 3 \times \sqrt{5}\).
The resulting expression \(3 \times \sqrt{5}\) is simplified as much as possible because you can't simplify \(\sqrt{5}\) any further. This process highlights the importance of recognizing perfect squares and using properties of square roots effectively.

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

If possible, simplify each radical expression. Assume that all variables represent positive real numbers. $$\sqrt[3]{125}$$

Simplify each expression, assuming that all variables represent nonnegative real numbers. $$\frac{1}{\sqrt{3}}-\frac{2}{\sqrt{12}}+2 \sqrt{3}$$

If possible, simplify each radical expression. Assume that all variables represent positive real numbers. $$\sqrt{24 m^{6} n^{5}}$$

If possible, simplify each radical expression. Assume that all variables represent positive real numbers. $$\sqrt{\frac{x^{5} y^{3}}{z^{2}}}$$

If possible, simplify each radical expression. Assume that all variables represent positive real numbers. $$\sqrt[4]{\frac{32 x^{5}}{y^{5}}}$$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

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.