Understanding Mean Proportional Calculation

The question asks us to find the mean proportional between two given numbers: $(21 + 16\sqrt{2})$ and $(16 – 7\sqrt{2})$.

The mean proportional, also known as the geometric mean, between two numbers $a$ and $b$ is given by the formula: $$ \text{Mean Proportional} = \sqrt{a \times b} $$

In this problem, let:

• $a = (21 + 16\sqrt{2})$
• $b = (16 – 7\sqrt{2})$

Calculating the Product of the Numbers

First, we need to calculate the product of $a$ and $b$: $$ a \times b = (21 + 16\sqrt{2}) \times (16 – 7\sqrt{2}) $$

We can expand this product using the distributive property (often remembered by the acronym FOIL):

• First terms: $21 \times 16 = 336$
• Outer terms: $21 \times (-7\sqrt{2}) = -147\sqrt{2}$
• Inner terms: $(16\sqrt{2}) \times 16 = 256\sqrt{2}$
• Last terms: $(16\sqrt{2}) \times (-7\sqrt{2}) = 16 \times (-7) \times (\sqrt{2} \times \sqrt{2}) = -112 \times 2 = -224$

Now, sum these results:

$$ a \times b = 336 - 147\sqrt{2} + 256\sqrt{2} - 224 $$

Combine the constant terms and the terms with $\sqrt{2}$:

$$ a \times b = (336 - 224) + (-147 + 256)\sqrt{2} $$

Simplify the expression:

$$ a \times b = 112 + 109\sqrt{2} $$

Finding the Mean Proportional

The mean proportional is the square root of this product:

$$ \text{Mean Proportional} = \sqrt{112 + 109\sqrt{2}} $$

Comparing with Options

Let's compare our result with the given options:

1. $17\sqrt{7}$
2. $\sqrt{112 + 109\sqrt{2}}$
3. $\sqrt{109 + 112\sqrt{2}}$
4. $4\sqrt{7}$

Our calculated mean proportional, $\sqrt{112 + 109\sqrt{2}}$, exactly matches the second option.