If `a^(2) + b^(2) = 88` and `ab = 6, (a gt 0, b gt 0)` then what is the value of `(a^(3) + b^(3))` ?

Options

A. 980
B. 1180
C. 820
D. 1000

Answer

To solve the problem step by step, we will use the given equations and algebraic identities.

### Step 1: Use the identity for \( a^2 + b^2 \)
We know that:
\[ a^2 + b^2 = (a + b)^2 - 2ab \]
Given \( a^2 + b^2 = 88 \) and \( ab = 6 \), we can substitute these values into the identity.

### Step 2: Substitute the known values
Substituting the values into the identity:
\[ 88 = (a + b)^2 - 2 \cdot 6 \]
This simplifies to:
\[ 88 = (a + b)^2 - 12 \]

### Step 3: Rearrange the equation
Now, we can rearrange the equation to find \( (a + b)^2 \):
\[ (a + b)^2 = 88 + 12 \]
\[ (a + b)^2 = 100 \]

### Step 4: Take the square root
Taking the square root of both sides gives us:
\[ a + b = \sqrt{100} = 10 \]

### Step 5: Use the identity for \( a^3 + b^3 \)
We know the formula for the sum of cubes:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
We can also express \( a^2 + b^2 \) in terms of \( a + b \) and \( ab \):
\[ a^2 - ab + b^2 = (a^2 + b^2) - ab \]

### Step 6: Substitute known values
Substituting the known values:
\[ a^2 - ab + b^2 = 88 - 6 = 82 \]

### Step 7: Calculate \( a^3 + b^3 \)
Now we can substitute \( a + b \) and \( a^2 - ab + b^2 \) into the sum of cubes formula:
\[ a^3 + b^3 = (10)(82) \]
\[ a^3 + b^3 = 820 \]

### Final Answer
Thus, the value of \( a^3 + b^3 \) is \( 820 \).

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