Let's calculate the given expression step-by-step and classify the resulting sum.
We need to find the sum of two numbers:
[tex]$ \frac{5}{6} + \sqrt{91} $[/tex]
### Step 1: Calculate the fractional part
Firstly, let's determine the exact value of the fraction:
[tex]$ \frac{5}{6} $[/tex]
This can be computed as:
[tex]$ \frac{5}{6} \approx 0.8333333333333334 $[/tex]
### Step 2: Calculate the square root part
Next, we calculate the square root of 91:
[tex]$ \sqrt{91} $[/tex]
We find that:
[tex]$ \sqrt{91} \approx 9.539392014169456 $[/tex]
### Step 3: Sum the two values
Now, let's add the two calculated values together:
[tex]$ 0.8333333333333334 + 9.539392014169456 $[/tex]
Adding these gives us:
[tex]$ 0.8333333333333334 + 9.539392014169456 \approx 10.37272534750279 $[/tex]
### Step 4: Classify the resulting sum
To classify this sum, we need to determine whether the result is a rational or irrational number.
A rational number can be written as a fraction of two integers, and it either terminates or repeats in its decimal form. An irrational number, on the other hand, cannot be written as a simple fraction and has a non-terminating, non-repeating decimal expansion.
Looking at the sum:
[tex]$ 10.37272534750279 $[/tex]
The decimal expansion does not terminate nor does it repeat, indicating that it is an irrational number.
### Summary
The calculation yields:
[tex]$ \frac{5}{6} \approx 0.8333333333333334 $[/tex]
[tex]$ \sqrt{91} \approx 9.539392014169456 $[/tex]
[tex]$ \frac{5}{6} + \sqrt{91} \approx 10.37272534750279 $[/tex]
The sum of these values, 10.37272534750279, is classified as irrational.
