If `f(x, y, z)= x+ y- z`, then `f((15)_(8), (15)_(10), (15)_(16))` = ?

Options

A. 16
B. 7
C. 10
D. All of the above

Answer

To solve the problem, we need to evaluate the function \( f(x, y, z) = x + y - z \) using the values of \( x \), \( y \), and \( z \) given in different bases. Let's break it down step by step.

### Step 1: Convert \( (15)_8 \) to Decimal
To convert the number \( (15)_8 \) (which is in base 8) to decimal, we use the formula:
\[ \text{Decimal Value} = d_1 \times b^1 + d_0 \times b^0 \]
where \( d_1 \) and \( d_0 \) are the digits of the number and \( b \) is the base.

For \( (15)_8 \):
- \( d_1 = 1 \) (the leftmost digit)
- \( d_0 = 5 \) (the rightmost digit)
- \( b = 8 \)

Calculating:
\[ (15)_8 = 1 \times 8^1 + 5 \times 8^0 = 1 \times 8 + 5 \times 1 = 8 + 5 = 13 \]
So, \( (15)_8 = 13 \) in decimal.

### Step 2: Convert \( (15)_{10} \) to Decimal
Since \( (15)_{10} \) is already in decimal, we have:
\[ (15)_{10} = 15 \]

### Step 3: Convert \( (15)_{16} \) to Decimal
To convert \( (15)_{16} \) (which is in base 16) to decimal, we use the same formula:
For \( (15)_{16} \):
- \( d_1 = 1 \)
- \( d_0 = 5 \)
- \( b = 16 \)

Calculating:
\[ (15)_{16} = 1 \times 16^1 + 5 \times 16^0 = 1 \times 16 + 5 \times 1 = 16 + 5 = 21 \]
So, \( (15)_{16} = 21 \) in decimal.

### Step 4: Substitute Values into the Function
Now we have:
- \( x = (15)_8 = 13 \)
- \( y = (15)_{10} = 15 \)
- \( z = (15)_{16} = 21 \)

Substituting these values into the function \( f(x, y, z) \):
\[ f(13, 15, 21) = 13 + 15 - 21 \]

### Step 5: Calculate the Result
Now we perform the arithmetic:
\[ 13 + 15 = 28 \]
\[ 28 - 21 = 7 \]

Thus, the final answer is:
\[ f((15)_{8}, (15)_{10}, (15)_{16}) = 7 \]