Questions: =(√2+7)(1/2√2)−(√2−7)(1/2√2)/(√2+7)^2

Solution Steps

To solve the given expression, we can simplify it step by step. First, notice that the numerator is a difference of two products, which can be simplified by factoring out the common term \(\frac{1}{2 \sqrt{2}}\). Then, simplify the expression further by performing the arithmetic operations. Finally, evaluate the entire expression using Python.

The given expression is:

\[ \frac{(\sqrt{2}+7)\left(\frac{1}{2 \sqrt{2}}\right)-(\sqrt{2}-7)\left(\frac{1}{2 \sqrt{2}}\right)}{(\sqrt{2}+7)^{2}} \]

First, factor out the common term \(\frac{1}{2 \sqrt{2}}\) from the numerator:

\[ \frac{1}{2 \sqrt{2}} \left[ (\sqrt{2} + 7) - (\sqrt{2} - 7) \right] \]

Simplify the expression inside the brackets:

\[ (\sqrt{2} + 7) - (\sqrt{2} - 7) = \sqrt{2} + 7 - \sqrt{2} + 7 = 14 \]

Thus, the numerator becomes:

\[ \frac{1}{2 \sqrt{2}} \times 14 = \frac{14}{2 \sqrt{2}} \]

The denominator is:

\[ (\sqrt{2} + 7)^2 \]

Expanding this, we have:

\[ (\sqrt{2})^2 + 2 \times \sqrt{2} \times 7 + 7^2 = 2 + 14\sqrt{2} + 49 = 51 + 14\sqrt{2} \]

Now, evaluate the entire expression:

\[ \frac{\frac{14}{2 \sqrt{2}}}{51 + 14\sqrt{2}} \]

Simplifying further:

\[ \frac{7}{\sqrt{2} \times (51 + 14\sqrt{2})} \]

Using the Python output, the result is approximately:

\[ 0.06991 \]

Final Answer