Certainly! Let's work through the given expression step-by-step:
Given expression:
[tex]\[
\log_7 \left( \frac{44 \cdot \sqrt[3]{343}}{\sqrt{2.401}} \right)
\][/tex]
### Step 1: Simplifying \(\sqrt[3]{343}\)
First, we need to simplify \(\sqrt[3]{343}\). Since \(343 = 7^3\), we have:
[tex]\[
\sqrt[3]{343} = \sqrt[3]{7^3} = 7
\][/tex]
### Step 2: Evaluating \(\sqrt{2.401}\)
Next, we calculate the square root of \(2.401\):
[tex]\[
\sqrt{2.401} \approx 1.5495160534825059
\][/tex]
### Step 3: Simplifying the Numerator
The numerator of the fraction is \(44 \cdot \sqrt[3]{343}\):
[tex]\[
44 \cdot 7 = 308
\][/tex]
### Step 4: Simplifying the Denominator
The denominator is already calculated as:
[tex]\[
\sqrt{2.401} \approx 1.5495160534825059
\][/tex]
### Step 5: Simplifying the Fraction
Now, let's simplify the expression inside the logarithm:
[tex]\[
\frac{308}{1.5495160534825059} \approx 198.77173863915527
\][/tex]
### Step 6: Calculating the Logarithm
Finally, we calculate the logarithm base 7 of the simplified value:
[tex]\[
\log_7 (198.77173863915527) \approx 2.719630773765796
\][/tex]
So, the value of the given expression is:
[tex]\[
\log_7 \left( \frac{44 \cdot \sqrt[3]{343}}{\sqrt{2.401}} \right) \approx 2.719630773765796
\][/tex]
