If a = 16 and b= 5 . Then value of `((a^2 + b^2 + ab)/(a^3-b^3))` is -----

Options

A. 1/11
B. 1/19
C. (121)/(3971)
D. None of these

Answer

To solve the expression \(\frac{a^2 + b^2 + ab}{a^3 - b^3}\) given \(a = 16\) and \(b = 5\), we can follow these steps:

### Step 1: Substitute the values of \(a\) and \(b\)
We start by substituting the values of \(a\) and \(b\) into the expression:
\[ \frac{16^2 + 5^2 + 16 \cdot 5}{16^3 - 5^3} \]

### Step 2: Calculate \(a^2\), \(b^2\), and \(ab\)
Now we calculate each term in the numerator:
- \(16^2 = 256\)
- \(5^2 = 25\)
- \(16 \cdot 5 = 80\)

Adding these together:
\[ 256 + 25 + 80 = 361 \]

### Step 3: Calculate \(a^3\) and \(b^3\)
Next, we calculate \(a^3\) and \(b^3\) for the denominator:
- \(16^3 = 4096\)
- \(5^3 = 125\)

Now, subtract \(b^3\) from \(a^3\):
\[ 4096 - 125 = 3971 \]

### Step 4: Substitute back into the expression
Now we can substitute the calculated values back into the expression:
\[ \frac{361}{3971} \]

### Step 5: Simplify the expression
The expression \(\frac{361}{3971}\) cannot be simplified further, but we can also notice that \(3971\) can be expressed as \(16^3 - 5^3\) using the identity:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]
Thus, we can rewrite the expression as:
\[ \frac{a^2 + ab + b^2}{(a - b)(a^2 + ab + b^2)} = \frac{1}{a - b} \]

### Step 6: Calculate \(a - b\)
Now we find \(a - b\):
\[ 16 - 5 = 11 \]

### Step 7: Final result
Thus, the final value of the expression is:
\[ \frac{1}{11} \]

### Final Answer:
The value of \(\frac{a^2 + b^2 + ab}{a^3 - b^3}\) is \(\frac{1}{11}\).

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