If `a^(3) = 1 + 7 , 3^(3) = 1 + 7 + b, 4^(4) = 1 + 7 + c`, then the value of a + b + c is

Options

A. 77
B. 110
C. 58
D. 75

Answer

To solve the problem step by step, we will break down the equations and find the values of \( a \), \( b \), and \( c \).

### Step 1: Calculate \( a \)
We start with the equation:
\[ a^3 = 1 + 7 \]
Calculating the right side:
\[ 1 + 7 = 8 \]
Now, we find \( a \):
\[ a^3 = 8 \]
Taking the cube root of both sides:
\[ a = \sqrt[3]{8} = 2 \]

### Step 2: Calculate \( b \)
Next, we use the equation:
\[ 3^3 = 1 + 7 + b \]
Calculating \( 3^3 \):
\[ 3^3 = 27 \]
Now substituting into the equation:
\[ 27 = 1 + 7 + b \]
Calculating the right side:
\[ 1 + 7 = 8 \]
So we have:
\[ 27 = 8 + b \]
Now, solving for \( b \):
\[ b = 27 - 8 = 19 \]

### Step 3: Calculate \( c \)
Now we look at the equation:
\[ 4^4 = 1 + 7 + c \]
Calculating \( 4^4 \):
\[ 4^4 = 256 \]
Substituting into the equation:
\[ 256 = 1 + 7 + c \]
Calculating the right side:
\[ 1 + 7 = 8 \]
So we have:
\[ 256 = 8 + c \]
Now, solving for \( c \):
\[ c = 256 - 8 = 248 \]

### Step 4: Calculate \( a + b + c \)
Now we can find the total:
\[ a + b + c = 2 + 19 + 248 \]
Calculating this:
\[ a + b + c = 269 \]

### Final Answer
The value of \( a + b + c \) is:
\[ \boxed{269} \]