What is the value of ` (a^(2) + b^(2))/( a^(3) - b^(3))` when a + b = 8 and a - b= 2 ?

Options

A. `0.313`
B. `0.347`
C. `0.368`
D. `0.381`

Answer

To solve the expression \(\frac{a^2 + b^2}{a^3 - b^3}\) given the equations \(a + b = 8\) and \(a - b = 2\), we can follow these steps:

### Step 1: Solve for \(a\) and \(b\)
We have two equations:
1. \(a + b = 8\)
2. \(a - b = 2\)

To find the values of \(a\) and \(b\), we can add these two equations together:

\[ (a + b) + (a - b) = 8 + 2 \]

This simplifies to:

\[ 2a = 10 \]

Dividing both sides by 2 gives:

\[ a = 5 \]

Now, we can substitute \(a\) back into one of the original equations to find \(b\). Using \(a + b = 8\):

\[ 5 + b = 8 \]

Subtracting 5 from both sides gives:

\[ b = 3 \]

### Step 2: Calculate \(a^2 + b^2\)
Now that we have \(a = 5\) and \(b = 3\), we can calculate \(a^2 + b^2\):

\[ a^2 + b^2 = 5^2 + 3^2 = 25 + 9 = 34 \]

### Step 3: Calculate \(a^3 - b^3\)
Next, we calculate \(a^3 - b^3\):

\[ a^3 - b^3 = 5^3 - 3^3 = 125 - 27 = 98 \]

### Step 4: Substitute into the expression
Now we can substitute \(a^2 + b^2\) and \(a^3 - b^3\) into the expression:

\[ \frac{a^2 + b^2}{a^3 - b^3} = \frac{34}{98} \]

### Step 5: Simplify the fraction
We can simplify \(\frac{34}{98}\) by dividing both the numerator and the denominator by 2:

\[ \frac{34 \div 2}{98 \div 2} = \frac{17}{49} \]

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

\[ \frac{17}{49} \]