`(23)_(5)+(47)_(9)=(?)_(8)`

Options

A. 70
B. 35
C. 64
D. 18

Answer

To solve the problem `(23)_(5) + (47)_(9) = (?)_(8)`, we will follow these steps:

### Step 1: Convert (23)_(5) to Decimal
To convert the base 5 number `23` to decimal, we use the formula:
\[ \text{Decimal} = d_1 \times b^1 + d_0 \times b^0 \]
where \( d_1 \) and \( d_0 \) are the digits of the number and \( b \) is the base.

For `23` in base 5:
- \( d_1 = 2 \) (the leftmost digit)
- \( d_0 = 3 \) (the rightmost digit)
- \( b = 5 \)

Calculating:
\[ 2 \times 5^1 + 3 \times 5^0 = 2 \times 5 + 3 \times 1 = 10 + 3 = 13 \]
So, `(23)_(5)` in decimal is `13`.

### Step 2: Convert (47)_(9) to Decimal
Next, we convert the base 9 number `47` to decimal using the same formula.

For `47` in base 9:
- \( d_1 = 4 \)
- \( d_0 = 7 \)
- \( b = 9 \)

Calculating:
\[ 4 \times 9^1 + 7 \times 9^0 = 4 \times 9 + 7 \times 1 = 36 + 7 = 43 \]
So, `(47)_(9)` in decimal is `43`.

### Step 3: Add the Decimal Values
Now we add the two decimal values obtained:
\[ 13 + 43 = 56 \]

### Step 4: Convert the Sum (56) to Base 8
Finally, we need to convert `56` from decimal to base 8. We do this by dividing the number by 8 and keeping track of the remainders.

1. Divide `56` by `8`:
- \( 56 \div 8 = 7 \) with a remainder of `0`
2. Divide `7` by `8`:
- \( 7 \div 8 = 0 \) with a remainder of `7`

Now we read the remainders from bottom to top, which gives us `70`.

### Final Answer
Thus, the answer to the equation `(23)_(5) + (47)_(9) = (?)_(8)` is:
\[ (70)_{8} \]