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Chapter 11: Problem 51

Simplify as completely as possible. (Assume \(x \geq 0 .)\) $$\frac{6+4 \sqrt{3}}{2}$$

Short Answer

Expert verified
The simplified form is \(3 + 2 \sqrt{3}\).

Step by step solution

01

Distribute the Denominator

The given expression is \( \frac{6+4 \sqrt{3}}{2} \). To simplify, distribute the denominator to each term in the numerator. So, divide both 6 and \(4 \sqrt{3}\) by 2.
02

Simplify Each Term

Divide 6 by 2, resulting in 3. Then, divide \(4 \sqrt{3}\) by 2, resulting in \(2 \sqrt{3}\).
03

Combine the Simplified Terms

Combine the simplified terms together to get the final simplified expression. Thus, the expression simplifies to \(3 + 2 \sqrt{3}\).

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Key Concepts

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

Distributing Denominator
When you are presented with a fraction like \(\frac{6+4 \sqrt{3}}{2}\), you need to simplify it step by step. The first step in this process is to distribute the denominator to each term in the numerator. This means you need to divide every individual term in the numerator by the denominator.

For instance, in our given fraction: \(\frac{6+4 \sqrt{3}}{2}\), you will divide both \(6\) and \(4 \sqrt{3}\) by \(2\).

Here鈥檚 how it breaks down:
  • Divide \(6\) by \(2\) to get \(3\).
  • Divide \(4 \sqrt{3}\) by \(2\) to get \(2 \sqrt{3}\).
By distributing the denominator, you convert the fraction into simpler parts.
Simplifying Radicals
Radicals can often make problems look more complicated than they are. But simplifying radicals is a very handy skill. For example, in our scenario, we have \(4 \sqrt{3}\) which needs to be simplified.

When simplifying the term \(4 \sqrt{3}\), considering the context of distributing the denominator, you want to divide by \(2\), which gives you \(2 \sqrt{3}\).

Here鈥檚 the work once the denominator is distributed:
  • The initial term is \(4 \sqrt{3}\).
  • Divide it by \(2\): \(4 \sqrt{3} \/ 2 = 2 \sqrt{3}\).
Simplifying radical terms helps you get manageable, easily combinable expressions.
Combining Like Terms
Combining like terms is a fundamental skill in algebra. It involves adding or subtracting terms that have the same variable part.

In our simplified expression from the previous steps, we have obtained \(3 + 2 \sqrt{3}\). Here, \(3\) and \(2 \sqrt{3}\) are the terms. You notice these terms are not like terms because one has \( \sqrt{3} \) accompanying it and the other does not.

As such, nothing more can be done to combine them because they are in their simplest forms.

The final simplified expression is hence: \(3 + 2 \sqrt{3}\). By learning how to distribute denominators, simplify radicals, and combine like terms, any algebra problem becomes easier to tackle.

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Most popular questions from this chapter

In Exercises \(85-94\), solve the equations using the square root method. Round off your answers to the nearest hundredth. $$t^{2}=21$$

Use the method you think is the most appropriate to solve the given equation. Check your answers by using a different method. $$\frac{x}{x+1}=\frac{x+2}{x+3}$$

Sketch the graphs of \(y=x^{2}, y=\frac{1}{2} x^{2},\) and \(y=\frac{1}{3} x^{2}\) on the same coordinate system. How would you describe the effect the coefficients \(\frac{1}{2}\) and \(\frac{1}{3}\) have on the graph of \(y=x^{2} ?\)

Sketch the graph of the given equation. Find the intercepts; approximate to the nearest tenth where necessary. $$y=x^{2}-4 x+2$$

In Exercises \(65-74\), solve the given equation. For quadratic equations, choose either the factoring method or the square root method, whichever you think is the easier to use. $$\frac{1}{x}+\frac{6}{x+2}=2$$
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