To determine which statement is true for the repeating decimal [tex]\(5.\overline{78}\)[/tex], let's analyze its nature step-by-step.
### Step 1: Understanding the Notation
The notation [tex]\(5.\overline{78}\)[/tex] indicates that the digits 78 repeat indefinitely. Therefore,
[tex]\[ 5.\overline{78} = 5.7878787878\ldots \][/tex]
### Step 2: Setting Up an Equation
Let's denote this repeating decimal by [tex]\(x\)[/tex]. So,
[tex]\[ x = 5.7878787878\ldots \][/tex]
### Step 3: Eliminate the Repeating Part
To eliminate the repeating part, we multiply [tex]\(x\)[/tex] by 100 (since the repeating block, 78, is two digits long):
[tex]\[ 100x = 578.7878787878\ldots \][/tex]
### Step 4: Subtract the Original Equation
Now, subtract the original [tex]\(x\)[/tex] from this equation:
[tex]\[ 100x - x = 578.7878787878\ldots - 5.7878787878\ldots \][/tex]
This simplifies to:
[tex]\[ 99x = 573 \][/tex]
### Step 5: Solve for [tex]\(x\)[/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[ x = \frac{573}{99} \][/tex]
### Step 6: Simplify the Fraction
Next, simplify the fraction [tex]\(\frac{573}{99}\)[/tex]:
Both 573 and 99 can be divided by 3:
[tex]\[ \frac{573 \div 3}{99 \div 3} = \frac{191}{33} \][/tex]
So,
[tex]\[ 5.\overline{78} = \frac{191}{33} \][/tex]
### Step 7: Classifying the Number
A number that can be expressed as a fraction of two integers (where the denominator is not zero) is a rational number. Since [tex]\(5.\overline{78}\)[/tex] can be expressed as [tex]\(\frac{191}{33}\)[/tex], it is a rational number.
### Conclusion
Given this analysis, the correct statement is:
A) It's rational because [tex]\(5.787878\ldots = \frac{191}{33}\)[/tex].
Therefore, we can confidently say that the valid choice is:
A) It's rational because [tex]\(5.787878\ldots = \frac{191}{33}\)[/tex].
