The value of `(19^(3)+6^(3)+(3)(19)(6)(25))/(3^(6)+6(243)(2)+(15)(81)(4)+(20)(27)(8)+(15)(9)(16)+(6)(3)(32)+2^(6))` is

Options

A. 19
B. 6
C. 25
D. 1

Answer

To solve the problem, we need to simplify the expression given:

\[ \frac{19^3 + 6^3 + 3 \cdot 19 \cdot 6 \cdot 25}{3^6 + 6 \cdot 243 \cdot 2 + 15 \cdot 81 \cdot 4 + 20 \cdot 27 \cdot 8 + 15 \cdot 9 \cdot 16 + 6 \cdot 3 \cdot 32 + 2^6} \]

### Step 1: Simplify the Numerator

The numerator is \(19^3 + 6^3 + 3 \cdot 19 \cdot 6 \cdot 25\). We can recognize that this can be expressed in the form of the sum of cubes:

\[ a^3 + b^3 + 3ab(a + b) = (a + b)^3 \]

where \(a = 19\) and \(b = 6\).

Calculating \(a + b\):

\[ a + b = 19 + 6 = 25 \]

Now, substituting \(a\) and \(b\) into the formula:

\[ 19^3 + 6^3 + 3 \cdot 19 \cdot 6 \cdot 25 = (19 + 6)^3 = 25^3 \]

So, the numerator simplifies to:

\[ 25^3 \]

### Step 2: Simplify the Denominator

Now, we need to simplify the denominator:

\[ 3^6 + 6 \cdot 243 \cdot 2 + 15 \cdot 81 \cdot 4 + 20 \cdot 27 \cdot 8 + 15 \cdot 9 \cdot 16 + 6 \cdot 3 \cdot 32 + 2^6 \]

Calculating each term separately:

1. \(3^6 = 729\)
2. \(6 \cdot 243 \cdot 2 = 2916\)
3. \(15 \cdot 81 \cdot 4 = 4860\)
4. \(20 \cdot 27 \cdot 8 = 4320\)
5. \(15 \cdot 9 \cdot 16 = 2160\)
6. \(6 \cdot 3 \cdot 32 = 576\)
7. \(2^6 = 64\)

Now, adding these values together:

\[ 729 + 2916 + 4860 + 4320 + 2160 + 576 + 64 = 15325 \]

### Step 3: Final Calculation

Now we can substitute back into the original expression:

\[ \frac{25^3}{15325} \]

Calculating \(25^3\):

\[ 25^3 = 15625 \]

Now substituting this into the fraction:

\[ \frac{15625}{15325} \]

### Step 4: Simplify the Fraction

To simplify \(\frac{15625}{15325}\), we can divide both the numerator and the denominator by 25:

\[ \frac{15625 \div 25}{15325 \div 25} = \frac{625}{613} \]

### Conclusion

Thus, the value of the given expression is:

\[ \frac{625}{613} \]