/*! 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 7 Simplify each radical (if possib... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 7

Simplify each radical (if possible). If imaginary, rewrite in terms of \(i\) and simplify. a. \(\sqrt{-16}\) b. \(\sqrt{-49}\) c. \(\sqrt{27}\) d. \(\sqrt{72}\)

Short Answer

Expert verified
a. \(4i\); b. \(7i\); c. \(3\sqrt{3}\); d. \(6\sqrt{2}\)

Step by step solution

01

Simplify \(\sqrt{-16}\)

The expression \(\sqrt{-16}\) is an imaginary number because we cannot take the square root of a negative number in the real number system. We rewrite \(\sqrt{-16}\) using \(i\), where \(i = \sqrt{-1}\). Therefore, \(\sqrt{-16} = \sqrt{16} \cdot \sqrt{-1} = 4i\).
02

Simplify \(\sqrt{-49}\)

Similar to the previous step, \(\sqrt{-49}\) is an imaginary number. We rewrite it with \(i\). So, \(\sqrt{-49} = \sqrt{49} \cdot \sqrt{-1} = 7i\).
03

Simplify \(\sqrt{27}\)

To simplify \(\sqrt{27}\), we look for perfect square factors of 27. Since 27 = 9 \(\times\) 3, and 9 is a perfect square, we have \(\sqrt{27} = \sqrt{9 \times 3} = \sqrt{9} \cdot \sqrt{3} = 3\sqrt{3}\).
04

Simplify \(\sqrt{72}\)

For \(\sqrt{72}\), we find the perfect square factors of 72. Since 72 = 36 \(\times\) 2, and 36 is a perfect square, we have \(\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \cdot \sqrt{2} = 6\sqrt{2}\).

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.

Simplifying Radicals
Simplifying radicals involves breaking down a radical, such as the square root (\(\sqrt{\cdots}\)), into its simplest form. This helps in making calculations easier and in providing a clearer understanding of the value the radical represents. The process often involves finding a perfect square factor of the number under the radical sign.
  • For instance, simplifying \(\sqrt{27}\) involves recognizing 9 as a perfect square factor of 27.
  • By rewriting 27 as \(9 \times 3\), it becomes possible to take the square root of 9 out of the radical, resulting in \(3\sqrt{3}\).
  • This step-by-step breakdown transforms a more complex radical into a combination of a whole number and a simpler radical expression.
Learning to simplify radicals is fundamental in mathematics, allowing for more straightforward operations and comparisons involving radicals.
Square Root of Negative Numbers
When we encounter the square root of a negative number, we delve into the realm of imaginary numbers. In the real number system, taking the square root of a negative number is not possible. This is where the unit imaginary number, \(i\), comes to the rescue.
  • The symbol \(i\) is defined as \(i = \sqrt{-1}\).
  • Using this definition, the square root of any negative number can be expressed as a real number multiplied by \(i\).
  • For example, \(\sqrt{-16}\) can be rewritten as \(\sqrt{16} \times \sqrt{-1} = 4i\).
Imaginary numbers expand our number system, allowing us to solve equations and problems involving negative square roots.
Complex Numbers
Complex numbers are an extension of our number system that includes both real and imaginary components. A complex number takes the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part.
  • Real part \(a\) is a regular number you are familiar with on the number line.
  • Imaginary part \(bi\) includes the unit \(i\), where \(i^2 = -1\).
  • Complex numbers often arise from operations involving the square roots of negative numbers or when working with polynomial equations.
By combining real and imaginary parts, complex numbers provide a comprehensive way to handle number operations that would be otherwise impossible in the real number system. Understanding complex numbers is crucial when dealing with all forms of advanced algebra.
Perfect Square Factors
To simplify radicals efficiently, identifying perfect square factors is a crucial step. A perfect square is any number that can be expressed as the square of an integer. Recognizing these factors allows for the extraction of terms out of the square root, simplifying calculations.
  • For \(\sqrt{72}\), identifying 36 as a perfect square factor helps because \(36 = 6^2\).
  • We can rewrite \(72\) as \(36 \times 2\), simplifying \(\sqrt{72}\) to \(6\sqrt{2}\).
  • Perfect square factors make radical simplification more manageable by reducing the radical part to its simplest form.
By skillfully identifying and using perfect square factors, students can streamline their process for working with radicals, leading to more precise and simplified results.

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

Fill in each blank with the appropriate word or phrase. Carefully reread the section, if necessary. "False solutions" to a rational or radical equation are also called ______ roots.

The time \(T\) (in days) for a planet to make one revolution around the sun is modeled by \(T=0.407 R^{\frac{3}{2}},\) where \(R\) is the maximum radius of the planet's orbit in millions of miles (Kepler's third law of planetary motion). Use the equation to approximate the maximum radius of each orbit, given the number of days it takes for one revolution. (See Appendix I.F, Exercises 45 and \(46 .\) ) a. Mercury: 88 days b. Venus: 225 days c. Earth: 365 days d. Mars: 687 days e. Jupiter: 4,333 days f. Saturn: 10,759 days

Identify the following equations as an identity, a contradiction, or a conditional equation, then state the solution. $$-3(4 z+5)=-15 z-20+3 z$$

Complex polynomials: Many techniques applied to solve polynomial equations with real coefficients can be applied to solve polynomial equations with complex coefficients. Here we apply the idea to carefully chosen quadratic equations, as a more general application must wait until a future course, when the square root of a complex number is fully developed. Solve each equation using the quadratic formula, noting that \(\frac{1}{i}=-i\). $$0.5 z^{2}+(4-3 i) z+(-9-12 i)=0$$

Solve using the zero product property. Be sure each equation is in standard form and factor out any common factors before attempting to solve. Check all answers in the original equation. $$22 x=x^{3}-9 x^{2}$$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

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.