/*! 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 18 Combine radicals, if possible. \... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 18

Combine radicals, if possible. \(5 \sqrt{9}-2 \sqrt{144}\)

Short Answer

Expert verified
Answer: The result is -9.

Step by step solution

01

Simplify the first radical

To simplify the first radical \(5\sqrt{9}\), find the square root of 9, which is 3. Multiply this by the coefficient 5 to obtain the simplified radical: \(5 \cdot 3 = 15\).
02

Simplify the second radical

To simplify the second radical \(-2\sqrt{144}\), find the square root of 144, which is 12. Multiply this by the coefficient -2 to obtain the simplified radical: \(-2 \cdot 12 = -24\).
03

Combine the simplified radicals

Now that we have simplified both radicals, we can combine them to find the final answer: \(15 + (-24) = -9\).

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 can often make the problem easier to solve. A radical is an expression that includes a root, such as a square root. To simplify radicals, you need to find the prime factorization of the number under the root. In simpler exercises, like our example, this can mean taking the square root of perfect squares.
For instance, in the expression \(5\sqrt{9} - 2\sqrt{144}\), we first simplify each radical separately.
  • Find the square root of 9, which gives us 3.
  • Multiply this by 5, the coefficient outside the radical, to simplify \(5\sqrt{9}\) to 15.
  • Similarly, \(\sqrt{144}\) is 12,
  • and multiplying it by -2 simplifies \(-2\sqrt{144}\) to -24.
These steps make it easier to work with radicals by reducing them to whole numbers.
Square Roots
Understanding square roots is key to simplifying and combining radicals. A square root asks "what number, when multiplied by itself, gives the original number?" Finding square roots is straightforward with perfect squares, like 9 and 144.
  • If you remember that 3 times 3 equals 9, then \(\sqrt{9} = 3\).
  • In the same way, \(12 \times 12\) results in 144, so \(\sqrt{144} = 12\).
These calculations are the first step when simplifying radicals. Recognizing perfect squares makes the process much quicker and helps in efficiently solving problems involving radicals. Practice helps in confidently spotting these patterns!
Arithmetic Operations with Radicals
After sorting out the square roots, the next step often involves performing arithmetic operations like addition or subtraction. However, it's important to note that you can only directly combine like radicals. Like radicals mean they have the same root and radicand.
In our original exercise, once you simplify:\(5\sqrt{9} = 15\) and \(-2\sqrt{144} = -24\), you can then perform straightforward arithmetic:
  • Add or subtract the simplified coefficients - here it would be 15 and -24.
  • The simplified radicals have different radicands, so we treat them separately in the operation step.
The expression simplifies to \(15 + (-24)\), which equals \(-9\). Remember, arithmetic with unlike radicals typically involves working through simplifying first, then combining the like terms if possible.

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

Write each expression without parentheses. Assume all variables are positive. $$ 3\left(2^{x} e^{x}\right)^{4} $$

By giving specific values for \(a, b,\) and \(c,\) explain how the exponent rule $$ \left(a^{b}\right)^{c}=a^{b c} $$ is used to rewrite the expressions $$ \sqrt{(x+1)\left(x^{2}+2 x+1\right)}=(x+1)^{3 / 2} $$

Write each expression without parentheses. Assume all variables are positive. $$ \left(\frac{6 g^{5}}{7 h^{7}}\right)^{2} $$

Write each expression as a product or a quotient. Assume all variables are positive. $$ 3^{2+3} $$

Rewrite each expression by rationalizing the denominator. $$ \frac{\sqrt{3}}{3 \sqrt{2}+\sqrt{3}} $$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Probability and 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.