If ` a = 11 and b = 9 ` then the value of `(( a^(2) + b^(2) + ab)/( a^(3) - b^(3))) ` is

Options

A. `(1)/(2)`
B. 2
C. `(1)/(20)`
D. 20

Answer

To solve the expression \(\frac{a^2 + b^2 + ab}{a^3 - b^3}\) given that \(a = 11\) and \(b = 9\), 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.

\[ a = 11, \quad b = 9 \]

### Step 2: Calculate \(a^2\), \(b^2\), and \(ab\)
Now, we calculate \(a^2\), \(b^2\), and \(ab\):

\[ a^2 = 11^2 = 121 \]
\[ b^2 = 9^2 = 81 \]
\[ ab = 11 \times 9 = 99 \]

### Step 3: Calculate \(a^2 + b^2 + ab\)
Next, we add these values together:

\[ a^2 + b^2 + ab = 121 + 81 + 99 \]

Calculating this gives:

\[ 121 + 81 = 202 \]
\[ 202 + 99 = 301 \]

So, \(a^2 + b^2 + ab = 301\).

### Step 4: Calculate \(a^3 - b^3\)
Now, we need to calculate \(a^3 - b^3\). We can use the formula for the difference of cubes:

\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]

First, we find \(a - b\):

\[ a - b = 11 - 9 = 2 \]

We already calculated \(a^2 + ab + b^2 = 301\). Now we can substitute these values into the formula:

\[ a^3 - b^3 = (2)(301) = 602 \]

### Step 5: Substitute back into the original expression
Now we can substitute back into the original expression:

\[ \frac{a^2 + b^2 + ab}{a^3 - b^3} = \frac{301}{602} \]

### Step 6: Simplify the fraction
We can simplify \(\frac{301}{602}\):

\[ \frac{301}{602} = \frac{1}{2} \]

### Final Answer
Thus, the value of \(\frac{a^2 + b^2 + ab}{a^3 - b^3}\) is \(\frac{1}{2}\).

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