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Chapter 10: Problem 35

Add or subtract. \(\frac{4 \sqrt{3}}{3}-\frac{\sqrt{12}}{3}\)

Short Answer

Expert verified
\(\frac{2\sqrt{3}}{3}\)

Step by step solution

01

Simplify Square Roots

First, simplify the square root in the expression \(\frac{\sqrt{12}}{3}\). We know \(\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}\). So, substitute back into the expression: \(\frac{4\sqrt{3}}{3} - \frac{2\sqrt{3}}{3}\).
02

Combine Like Terms

Since both terms \(\frac{4\sqrt{3}}{3}\) and \(\frac{2\sqrt{3}}{3}\) are like terms, with the same denominator, we can combine them by subtracting their numerators: \(\frac{(4 - 2)\sqrt{3}}{3} = \frac{2\sqrt{3}}{3}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simplifying Square Roots
Square roots can seem intimidating at first, but simplifying them can often make the math clearer. When you have a square root like \( \sqrt{12} \), try to express it in terms of smaller square roots that you know. Think about the factors of 12: it can be broken down into 4 and 3, which gives us \( \sqrt{4 \times 3} \). Because \( \sqrt{4} = 2 \), this expression can then be simplified to \( 2\sqrt{3} \). So, remember:
  • Find factors of the number under the square root.
  • Identify perfect squares among these factors.
  • Simplify the expression using known square roots.
With practice, simplifying square roots becomes a quick and natural part of solving these problems.
Rational Expressions
A rational expression is simply a fraction where the numerator and the denominator contain algebraic expressions. In our exercise, we have expressions with square roots in the numerator and a regular number in the denominator, like \( \frac{4\sqrt{3}}{3} \). The goal with rational expressions is to combine them or simplify them as much as possible.Key steps for handling rational expressions include:
  • Simplifying components like square roots if possible.
  • Finding common denominators when combining fractions.
  • Reducing the fraction, if possible, by dividing both the numerator and the denominator by their greatest common factor.
By following these steps, rational expressions can be handled with less hassle, leading to more efficient problem solving.
Combining Like Terms
Combining like terms is an important skill in simplifying algebraic expressions. The idea is to group and combine terms that have the same variable or radical parts. In the given expression, \( \frac{4\sqrt{3}}{3} - \frac{2\sqrt{3}}{3} \), both terms contain the same square root, \( \sqrt{3} \), and share the same denominator, 3.Here's how to combine them efficiently:
  • Ensure the terms have the same denominator, which supports straightforward addition or subtraction.
  • Look at the numerators; since they both involve \( \sqrt{3} \), we only need to subtract the coefficients of \( \sqrt{3} \), 4 and 2.
  • Subtract these coefficients: \( 4 - 2 = 2 \). Our result is \( \frac{2\sqrt{3}}{3} \).
Mastering how to combine like terms helps simplify complex algebraic expressions with ease, making the overall process more approachable and understandable.

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Most popular questions from this chapter

Identify the domain and then graph each function. $$f(x)=\sqrt{x+1}$$ use the following table. $$\begin{array}{|c|c|} \hline x & {f(x)} \\ \hline-1 & {} \\ \hline 0 & {} \\ \hline 3 & {} \\ \hline 8 & {} \\ \hline \end{array}$$

Write each integer as a product of two integers such that one of the factors is a perfect square. For example, write 18 as \(9.2,\) because 9 is a perfect square. $$ 48 $$

If \(f(x)=\sqrt{2 x+3}\) and \(g(x)=\sqrt[3]{x-8},\) find the following function values. $$f(2)$$

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The number of cellular telephone subscriptions in the United States from 1996 through 2006 can be modeled by the function \(f(x)=33.3 x^{45},\) where \(y\) is the number of cellular telephone subscriptions in millions, \(x\) years after \(1996 .\) (Source: Based on data from the Cellular Telecommunications \& Internet Association, \(1994-2000)\) Use this information to answer Exercises. Use this model to estimate the number of cellular telephone subscriptions in the United States in \(2006 .\) Round to the nearest tenth of a million.
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