The mean proportional between $234$ and $104$ is

### Concept & Formula The mean proportional $x$ between two numbers $a$ and $b$ is the geometric mean of the numbers, calculated as: $$ x = \sqrt{a \times b} $$ ### Step-by-Step Solution 1. Let $a = 234$ and $b = 104$. The mean proportional is $x = \sqrt{234 \times 104}$. 2. To avoid large multiplications, find the prime factorization of each number. $234 = 2 \times 117 = 2 \times 9 \times 13 = 2 \times 3^2 \times 13$ $104 = 8 \times 13 = 2^3 \times 13$ 3. Multiply the factorizations inside the square root: $$ x = \sqrt{(2 \times 3^2 \times 13) \times (2^3 \times 13)} $$ 4. Combine powers of the same base: $$ x = \sqrt{2^4 \times 3^2 \times 13^2} $$ 5. Take the square root by halving the exponents: $x = 2^2 \times 3^1 \times 13^1 = 4 \times 3 \times 13$ 6. Calculate the final value: $x = 12 \times 13 = 156$ 7. Since 156 is not among the specific options $12$, $39$, or $54$, the answer is "None of these". ### Exam Strategy & Shortcut Look at the unit digits. $234 \times 104$ ends in $4 \times 4 = 16$, so the product ends in 6. The square root of a number ending in 6 must end in 4 or 6. The given numerical options are 12 (ends in 2, square ends in 4), 39 (ends in 9, square ends in 1), and 54 (ends in 4, square ends in 6). We can quickly check $54^2 \approx 2500$, but $234 \times 104$ is roughly $234 \times 100 \approx 23400$. The options are way too small. ### Common Pitfall Multiplying 234 by 104 to get 24336 and then struggling to find the square root of a large 5-digit number without a calculator. Always factorize first. ### Final Answer Therefore, the correct answer is **None of these**.