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Chapter 4: Problem 29

Simplify the expression. $$\frac{5}{9}+\frac{5}{9} \cdot \frac{7}{9}$$

Short Answer

Expert verified
The simplified expression is \( \frac{80}{81} \).

Step by step solution

01

Simplify the Multiplication

First, multiply the fraction \( \frac{5}{9} \) by \( \frac{7}{9} \). Multiply the numerators together and the denominators together: \[ \frac{5 \cdot 7}{9 \cdot 9} = \frac{35}{81} \]. So, \( \frac{5}{9} \cdot \frac{7}{9} = \frac{35}{81} \).
02

Add the Fractions

Next, add the fraction \( \frac{5}{9} \) to \( \frac{35}{81} \). The least common denominator is 81. Convert \( \frac{5}{9} \) to have the same denominator: \( \frac{5}{9} = \frac{5 \times 9}{9 \times 9} = \frac{45}{81} \). Now add the fractions: \[ \frac{45}{81} + \frac{35}{81} = \frac{45 + 35}{81} = \frac{80}{81} \].
03

Final Result

The final simplified form of the expression is \( \frac{80}{81} \). There are no common factors in the numerator and denominator, so the fraction is already in its simplest form.

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Key Concepts

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

Simplifying Fractions
Simplifying fractions is all about making a fraction as simple as possible. This means ensuring that the numerator (top number) and the denominator (bottom number) have no common factors other than 1.
To simplify a fraction, follow these simple steps:
  • Identify the greatest common factor (GCF) of both the numerator and the denominator.
  • Divide both the numerator and the denominator by this GCF.
This process ensures that your fraction is in its simplest terms, making it easier to work with in algebraic expressions and computations.
In our example, the final result was \( \frac{80}{81} \). Since 80 and 81 have no common factors other than 1, the fraction is already simplified. Always check divisibility rules by small primes like 2, 3, and 5 to confirm simplification.
Common Denominator
A common denominator is necessary when adding or subtracting fractions. It allows us to combine fractions by expressing them on a common scale.
To find a common denominator:
  • Look for the least common multiple (LCM) of the denominators.
  • Multiply the numerators and denominators to create equivalent fractions with this common denominator.
In the exercise above, the fractions \( \frac{5}{9} \) and \( \frac{35}{81} \) were involved. While adding them, we noticed \( 81 \) as the least common denominator.
This allowed us to convert \( \frac{5}{9} \) to \( \frac{45}{81} \), facilitating straightforward addition. A common denominator helps unify the fractions seamlessly, enabling accurate computations.
Fraction Addition
Adding fractions requires a clear understanding of their common denominators. Once fractions share a denominator, you only need to add their numerators while keeping the denominator the same.
Here's a quick guide to fraction addition:
  • Ensure fractions have a common denominator.
  • Add the numerators together.
  • Keep the denominator as it is.
  • Simplify the resultant fraction if necessary.
As demonstrated, once \( \frac{5}{9} \) was converted to \( \frac{45}{81} \), we simply said \( \frac{45}{81} + \frac{35}{81} = \frac{80}{81} \). This illustrates how straightforward fraction addition becomes when processing a uniform denominator. Nonetheless, always strive to simplify the result when possible, ensuring your answer is as concise as it can be.

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Most popular questions from this chapter

Solve the equation and simplify your answer. $$-6 x=-9 x+8$$

Solve the equation and simplify your answer. $$-\frac{3}{2} x=\frac{8}{7}$$

Is \(2 / 7\) a solution of the equation \(x-\frac{4}{9}=-\frac{8}{63}\) ?

Solve the equation and simplify your answer. $$-\frac{4}{9} x-\frac{8}{9}=\frac{1}{2} x-\frac{1}{2}$$

Is \(3 / 8\) a solution of the equation \(x-\frac{5}{9}=-\frac{13}{72}\) ?
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