`(52)_(7)+46_(8)=(?)_(10)`

Options

A. `(75)_(10)`
B. `(50)_(10)`
C. `(39)_(39)`
D. `(28)_(10)`

Answer

To solve the problem `(52)_(7) + (46)_(8) = (?)_(10)`, we will convert both numbers from their respective bases to base 10, perform the addition, and then express the result in base 10.

### Step 1: Convert (52)_(7) to base 10
The number (52) in base 7 can be converted to base 10 using the formula:
\[ \text{Value} = a \cdot x^1 + b \cdot x^0 \]
where \( a \) and \( b \) are the digits of the number and \( x \) is the base.

For (52)_(7):
- \( a = 5 \)
- \( b = 2 \)
- \( x = 7 \)

Calculating:
\[ 5 \cdot 7^1 + 2 \cdot 7^0 = 5 \cdot 7 + 2 \cdot 1 = 35 + 2 = 37 \]

### Step 2: Convert (46)_(8) to base 10
Now, we convert (46) in base 8 to base 10 using the same formula.

For (46)_(8):
- \( a = 4 \)
- \( b = 6 \)
- \( x = 8 \)

Calculating:
\[ 4 \cdot 8^1 + 6 \cdot 8^0 = 4 \cdot 8 + 6 \cdot 1 = 32 + 6 = 38 \]

### Step 3: Add the two base 10 values
Now, we add the two results from steps 1 and 2:
\[ 37 + 38 = 75 \]

### Step 4: Write the final result in base 10
The result of the addition is \( 75 \) in base 10.

Thus, the final answer is:
\[ (52)_{7} + (46)_{8} = (75)_{10} \]

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