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Chapter 10: Problem 15

Add or subtract. \(\frac{\sqrt{20 x}}{9}+\sqrt{\frac{5 x}{9}}\)

Short Answer

Expert verified
The result is \( \frac{5\sqrt{5x}}{9} \).

Step by step solution

01

Simplify Square Roots

Start by simplifying the square roots in each expression. For the first term, \( \sqrt{20x} \) can be written as \( \sqrt{4 \times 5x} \) which simplifies to \( 2\sqrt{5x} \). The second term involves \( \sqrt{\frac{5x}{9}} \), which can be written as \( \sqrt{\frac{5x}{9}} = \frac{\sqrt{5x}}{3} \). This is because \( \sqrt{\frac{5x}{9}} = \sqrt{\frac{1}{9}} \times \sqrt{5x} \) and \( \sqrt{\frac{1}{9}} = \frac{1}{3} \). Thus, the terms become \( \frac{2\sqrt{5x}}{9} \) and \( \frac{\sqrt{5x}}{3} \).
02

Find a Common Denominator

To add these two fractions, we need a common denominator. The two denominators are 9 and 3. The least common denominator is 9. Rewrite \( \frac{\sqrt{5x}}{3} \) with the common denominator 9: \( \frac{\sqrt{5x}}{3} = \frac{\sqrt{5x} \times 3}{3 \times 3} = \frac{3\sqrt{5x}}{9} \).
03

Add Fractions

Now that both fractions have the same denominator, add them: \( \frac{2\sqrt{5x}}{9} + \frac{3\sqrt{5x}}{9} = \frac{(2\sqrt{5x} + 3\sqrt{5x})}{9} = \frac{5\sqrt{5x}}{9} \).
04

Simplify the Expression

The final expression \( \frac{5\sqrt{5x}}{9} \) is already in its simplest form, and no further simplification is possible.

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Key Concepts

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

Simplifying Square Roots
Simplifying square roots is a key skill in algebra. The process involves expressing the square root of a number or variable in its simplest form.
  • To simplify a square root, find the largest perfect square factor of the number or expression under the square root.
  • For example, with the expression \( \sqrt{20x} \), recognize that 20 can be factored into \( 4 \times 5 \), where 4 is a perfect square.
  • This allows us to rewrite \( \sqrt{20x} \) as \( \sqrt{4 \times 5x} \), which simplifies further to \( 2\sqrt{5x} \).
  • A similar approach applies to fractional square roots, such as \( \sqrt{\frac{5x}{9}} \).
  • This can be expressed as \( \frac{\sqrt{5x}}{\sqrt{9}} \), and since \( \sqrt{9} = 3 \), it becomes \( \frac{\sqrt{5x}}{3} \).
Practicing such simplifications helps make algebraic manipulations much more manageable.
Common Denominator
When adding or subtracting fractions, finding a common denominator is important. It ensures that the fractions are compatible for direct addition or subtraction.
  • The common denominator is a shared multiple of the denominators of the fractions you are working with.
  • For the fractions \( \frac{2\sqrt{5x}}{9} \) and \( \frac{\sqrt{5x}}{3} \), notice the denominators: 9 and 3.
  • The least common denominator here is 9, since it is the smallest number that both denominators can divide into without leaving a remainder.
  • Adjust the second fraction by expressing \( \frac{\sqrt{5x}}{3} \) as \( \frac{3\sqrt{5x}}{9} \) to match the common denominator found.
Once the fractions share the same denominator, you can easily add or subtract them.
Adding Fractions
Once fractions have a common denominator, you can add or subtract their numerators directly.
  • Take the fractions \( \frac{2\sqrt{5x}}{9} \) and \( \frac{3\sqrt{5x}}{9} \) which both now have the denominator 9.
  • Add the numerators: \( 2\sqrt{5x} + 3\sqrt{5x} = 5\sqrt{5x} \).
  • The result is then written over the common denominator: \( \frac{5\sqrt{5x}}{9} \).
  • This expression is the sum of the two fractions, simplified and combined.
Addition of fractions becomes straightforward once you attain a clear common denominator鈥攅nhancing clarity and reducing complexity in the solution process.

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