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Chapter 9: Problem 90

Rewrite the expression as a single fraction and simplify. $$\frac{2}{\sqrt{5}}-9$$

Short Answer

Expert verified
The expression simplified as a single fraction is \(\frac{2\sqrt{5} + 45}{5}\).

Step by step solution

01

Rationalize the Denominator of the First Term

To rationalize the denominator of the fraction \(\frac{2}{\sqrt{5}}\), multiply both the numerator and the denominator by \(\sqrt{5}\). This simplifies the expression to \(\frac{2\sqrt{5}}{5}\).
02

Rewrite the Second Term as a Fraction

Rewrite \(-9\) as a fraction with denominator 1, i.e., \(-9\) is equivalent to \(\frac{-9}{1}\).
03

Find a Common Denominator

The common denominator between 5 and 1 is 5. Thus, the second term is multiplied top and bottom by 5, to get \(\frac{-45}{5}\).
04

Combine the Fractions

Combine the two fractions into a single fraction, \(\frac{2\sqrt{5}}{5} - \frac{-45}{5}\), by subtracting the second fraction from the first. This results in \(\frac{2\sqrt{5} + 45}{5}\).

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Key Concepts

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

Rationalizing Denominators
Rationalizing the denominator is a technique used to eliminate square roots from the bottom of a fraction. It helps to simplify expressions and makes them easier to work with. In math, having a radical in the denominator is less preferred. To rationalize the denominator, we multiply both the numerator and the denominator by a radical that will help simplify the square root.

For example, in the fraction \(\frac{2}{\sqrt{5}}\), the denominator is \(\sqrt{5}\). By multiplying both the top and the bottom by \(\sqrt{5}\), we have:
  • Numerator: \(2 \times \sqrt{5} = 2\sqrt{5}\)
  • Denominator: \(\sqrt{5} \times \sqrt{5} = 5\)
This results in \(\frac{2\sqrt{5}}{5}\), where the denominator is now a rational number, 5, which is cleaner.
Common Denominators
Fractions need common denominators to be added or subtracted together. A common denominator is a shared multiple of the denominators of two or more fractions. Finding this common denominator simplifies operations and helps you combine or compare the fractions easily.

When you have fractions like \(\frac{2\sqrt{5}}{5}\) and \(\frac{-9}{1}\), the next step is to bring them to a common denominator. Here, the denominators are 5 and 1 respectively. The least common denominator (LCD) of 5 and 1 is 5, since 5 is the smallest number that both denominators divide into without a remainder.
  • The first fraction remains \(\frac{2\sqrt{5}}{5}\)
  • Convert the second fraction by multiplying its numerator and denominator by 5: \(\frac{-9}{1} \times \frac{5}{5} = \frac{-45}{5}\)
Now, both fractions share a denominator of 5.
Combining Fractions
Once fractions have a common denominator, they can be combined into a single fraction. This is the final step to simplifying algebraic expressions involving subtraction or addition of fractions.

For our fractions \(\frac{2\sqrt{5}}{5}\) and \(\frac{-45}{5}\), combining them involves adding or, in this case, subtracting their numerators while keeping the common denominator:
  • Numerator: \(2\sqrt{5} - (-45) = 2\sqrt{5} + 45\)
  • Denominator remains as 5.
Thus, the combined and simplified expression is \(\frac{2\sqrt{5} + 45}{5}\). This process makes handling complex algebraic expressions with fractions more straightforward.

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Most popular questions from this chapter

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