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Chapter 8: Problem 76

Write each quotient in lowest terms. $$ \frac{25+\sqrt{75}}{10} $$

Short Answer

Expert verified
\(\frac{5 + \sqrt{3}}{2}\)

Step by step solution

01

Simplify the Square Root

Simplify \(\sqrt{75}\). \(75 = 25 \cdot 3\), so \(\sqrt{75} = \sqrt{25 \cdot 3} = \sqrt{25} \cdot \sqrt{3} = 5\sqrt{3}\).
02

Substitute Back

Replace \(\sqrt{75}\) with \(5\sqrt{3}\) in the original expression: \(\frac{25 + 5\sqrt{3}}{10}\).
03

Factor the Numerator

Factor out 5 from the numerator: \(\frac{5(5 + \sqrt{3})}{10}\).
04

Simplify the Fraction

Divide the numerator and the denominator by 5: \(\frac{5(5 + \sqrt{3})}{10} = \frac{5 + \sqrt{3}}{2}\).

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Key Concepts

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

square roots
A square root is a special value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4 because 4 times 4 equals 16.
For the exercise provided, we need to simplify \( \sqrt{75} \). We can do this by factoring the number inside the square root:
  • 75 can be written as 25 multiplied by 3
  • This means \( \sqrt{75} = \sqrt{25 \cdot 3} = \sqrt{25} \cdot \sqrt{3} \)
  • Since \( \sqrt{25} = 5, \) we end up with \( \sqrt{75} = 5 \sqrt{3} \)
Understanding how to simplify square roots is important because it helps us make expressions easier to work with.
factoring
Factoring is the process of breaking down a number or expression into its prime components. For example, if you have the number 12, you can factor it into 2 times 6, and then break it down further into 2 times 2 times 3.
In the given exercise, after we've simplified the square root, we have: \(\frac{25 + 5\sqrt{3}}{10}\). Next, we factor out the greatest common factor from the numerator.
  • Here, 5 can be factored out. So we rewrite the numerator as \( 5(5 + \sqrt{3}) \)
This makes our equation: \(\frac{5(5 + \sqrt{3})}{10}\).
Factoring helps simplify quotients and products, making it easier to perform arithmetic operations.
lowest terms
To simplify a fraction to its lowest terms, we divide the numerator and the denominator by their greatest common divisor (GCD). This step ensures the fraction is in its simplest form.
For the exercise problem, after factoring, we have the fraction \( \frac{5(5 + \sqrt{3})}{10} \). The numerator and denominator share a common factor of 5:
  • Divide both by 5: \( \frac{5(5 + \sqrt{3})}{10} = \frac{5 + \sqrt{3}}{2}\)
Now, the fraction is simplified to its lowest terms.
This process is crucial as it helps us present our final answers in the simplest form, making them easier to understand and work with.

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