Question

Find the mean proportional of (12 + 6√2) and (8 - 4√2).

• A. 4√3
• B. 3√2
• C. 3√3
• D. 4√2

Answer 1

Answer:

A. 4√3

The mean proportional (geometric mean) between two numbers (a) and (b) is:

$\sqrt{a \times b}$
Here,
$a = (12 + 6\sqrt{2}), \quad b = (8 - 4\sqrt{2})$
Multiply the two expressions

$(12 + 6\sqrt{2})(8 - 4\sqrt{2})$
Expand:
$= 12 \cdot 8 + 12 \cdot (-4\sqrt{2}) + 6\sqrt{2} \cdot 8 + 6\sqrt{2} \cdot (-4\sqrt{2})$
$= 96 - 48\sqrt{2} + 48\sqrt{2} - 24 \cdot 2$
$= 96 - 48 = 48$
Take square root

$\sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}$