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Chapter 8: Problem 71

Write each quotient in lowest terms. $$ \frac{2 \sqrt{3}+10}{8} $$

Short Answer

Expert verified
\( \frac{\sqrt{3}+5}{4} \)

Step by step solution

01

Identify Common Factors

Examine the terms in the numerator and the denominator to identify any common factors. In this case, the numerator is composed of two terms: \(2\sqrt{3}\) and 10, and the denominator is 8.
02

Factor Out the Greatest Common Factor (GCF)

The greatest common factor for the terms in the numerator (2\(\sqrt{3}\) and 10) and the denominator (8) is 2. So, factor out 2 from the numerator: \[ \frac{2(\sqrt{3}+5)}{8} \]
03

Simplify the Fraction

Divide both the numerator and the denominator by their greatest common factor, which is 2: \[ \frac{2(\sqrt{3}+5)}{8} = \frac{\sqrt{3}+5}{4} \]
04

Write the Final Result

After simplifying, the quotient in its lowest terms is: \[ \frac{\sqrt{3}+5}{4} \]

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Key Concepts

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

Greatest Common Factor
When simplifying algebraic expressions, identifying the Greatest Common Factor (GCF) is crucial. The GCF is the highest number that evenly divides all terms in an expression. In our example, we started with the fraction \(\frac{2\text{\textbackslash sqrt\textbraceleft 3\textbraceleft}+10}{8}\).
To identify the GCF of the numerator, we looked at the numbers 2 and 10. Both have a common factor of 2.
The denominator, 8, also has a common factor with the numerator, which is 2.
Always try to find the largest number that simplifies the entire fraction!
Fraction Simplification
Simplifying fractions is another essential skill in algebra. After factoring out the GCF in our example, we obtained the fraction \(\frac{2(\text{\textbackslash sqrt\textbraceleft 3\textbraceleft}+5)}{8}\).
Next, we divided both the numerator and the denominator by their GCF, which was 2.
This step brought us to the simplified fraction: \( \frac{\text{\textbackslash sqrt\textbraceleft 3\textbraceleft}+5}{4} \).
Remember, the goal of simplification is to get the fraction to its lowest terms, ensuring no more common factors exist between the numerator and the denominator.
Rational Expressions
Rational expressions are fractions where the numerator and/or the denominator are algebraic expressions. The example we worked on simplifies the rational expression \( \frac{2\text{\textbackslash sqrt\textbraceleft 3\textbraceleft}+10}{8} \).
Here are some tips when dealing with rational expressions:
  • Always look for common factors between the numerator and the denominator.
  • Simplify by dividing both by their GCF.
  • Check your final expression to ensure it's in the simplest form.
For our fraction, simplifying it resulted in \( \frac{\text{\textbackslash sqrt\textbraceleft 3\textbraceleft}+5}{4} \). Simplifying rational expressions often makes complex algebraic problems more manageable.

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Most popular questions from this chapter

If an investment of \(P\) dollars grows to \(A\) dollars in 2 yr, the annual rate of return on the investment is given by $$ r=\frac{\sqrt{A}-\sqrt{P}}{\sqrt{P}} $$ First rationalize the denominator, and then find the annual rate of return (as a percent) if \(\$ 50,000\) increases to \(\$ 54,080.\)

Simplify each radical. $$ \sqrt[3]{-\frac{216}{125}} $$

Find each product and simplify. Simplify the radical \(\sqrt{288}\) in two ways. First, factor 288 as \(144 \cdot 2\) and then simplify. Second, factor 288 as \(48 \cdot 6\) and then simplify. How do the answers compare? Make a conjecture concerning the quickest way to simplify such a radical.

Solve each equation. $$ (2 x-1)(4 x-3)=0 $$

Solve each problem. The square root of the sum of a number and 4 is \(5 .\) Find the number.
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