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Chapter 9: Problem 53

Perform the indicated operations and simplify. $$\frac{5}{6 x}+\frac{3}{8 x^{2}}$$

Short Answer

Expert verified
\frac{20x + 9}{24x^2}

Step by step solution

01

- Find a Common Denominator

The common denominator for the two fractions \(\frac{5}{6x}\) and \(\frac{3}{8x^2}\) is \((6x) \times (8x) = 48x^2\).
02

- Rewrite Each Fraction with the Common Denominator

Rewrite \(\frac{5}{6x}\) as \(\frac{5 \times 8x}{48x^2} = \frac{40x}{48x^2}\). Rewrite \(\frac{3}{8x^2}\) as \(\frac{3 \times 6}{48x^2} = \frac{18}{48x^2}\).
03

- Add the Fractions

Combine the fractions: \[ \frac{40x}{48x^2} + \frac{18}{48x^2} = \frac{40x + 18}{48x^2} \]
04

- Simplify the Result

Factor out the common factors in the numerator: \[ \frac{40x + 18}{48x^2} = \frac{2 \times (20x + 9)}{2 \times 24x^2} = \frac{20x + 9}{24x^2} \]

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Key Concepts

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

Common Denominator
When adding or subtracting fractions, finding a common denominator is crucial. The common denominator is a shared multiple of the denominators from all fractions involved. It allows us to combine the fractions correctly.

In our example, we have the fractions \(\frac{5}{6x}\) and \(\frac{3}{8x^2}\). To add these, we must find a common denominator. We go step-by-step:
  • Identify the denominators: 6x and 8x虏.
  • Find the Least Common Multiple (LCM) of these denominators.

For this, multiply the highest powers of all variables and constants: (6x) 脳 (8x) = 48x虏.
So, the common denominator for these fractions is 48x虏.
Simplifying Fractions
Simplifying fractions means reducing them to their simplest form. It's a process of dividing the numerator and the denominator by their greatest common divisor (GCD).

After finding a common denominator, rewrite each fraction. In our example:
  • Rewrite \(\frac{5}{6x}\) as \(\frac{5 脳 8x}{48x虏} = \frac{40x}{48x^2}\).
  • Rewrite \(\frac{3}{8x^2}\) as \(\frac{3 脳 6}{48x虏} = \frac{18}{48x虏}\).
Now, add the fractions:
\(\frac{40x}{48x^2} + \frac{18}{48x^2} = \frac{40x + 18}{48x^2} \).

Next is the simplification step. Factor out any common terms in the numerator and denominator:
\(\frac{2 脳 (20x + 9)}{2 脳 24x虏} = \frac{20x + 9}{24x^2}\).
Thus, the simplified form is \(\frac{20x + 9}{24x虏}\).
Algebraic Fractions
Algebraic fractions are fractions where the numerator, the denominator, or both contain algebraic expressions (variables, constants, or both). Handling these fractions largely involves the same operations as numerical fractions but with extra care to manage the variables correctly.

In our exercise, the given fractions are \(\frac{5}{6x}\) and \(\frac{3}{8x^2}\). These fractions include variables (x) in their denominators:
  • First, find a common denominator (48x虏 in this case).
  • Rewrite the fractions using this common denominator.
  • Combine the fractions by adding the new numerators over the common denominator.
  • Simplify the final fraction by factoring and reducing common elements.
Working with algebraic fractions often requires practice to become comfortable. Break down each step to make the process manageable, ensuring you clearly understand the role of each variable and constant involved.

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Most popular questions from this chapter

If the exercise is an equation, solve it and check. Otherwise, perform the indicated operations and simplify. $$\frac{6}{r^{2}-9}=\frac{3}{r^{2}-3 r}$$

Each of Exercises \(37-50\) is a formula either from mathematics or the physical or social sciences. Solve each of the formulas for the indicated variable. $$F=\frac{9}{5} C+32 \text { for } C$$

If the exercise is an equation, solve it and check. Otherwise, perform the indicated operations and simplify. $$\frac{1}{x+3} \div \frac{x+1}{x^{2}-9}$$

Each of Exercises \(37-50\) is a formula either from mathematics or the physical or social sciences. Solve each of the formulas for the indicated variable. $$S=S_{0}+v_{0} t+\frac{1}{2} g t^{2} \quad \text { for } g$$

Find the value of \(t\) so that the line passing through the points \((t, 4)\) and \((3, t)\) is parallel to the line passing through the points \((0,-1)\) and \((4,-3)\)
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