To solve the given expression, we will first evaluate the numerator and the denominator separately.

The given expression is:

[ \frac{15(21.086) - (514)(429)}{\sqrt{[15(26.550) - (514)^2] \times [15(26.191) - (429)^2]}} ]

Step 1: Evaluate the Numerator

$15 \times 21.086 - 514 \times 429$

First, calculate each part:

$15 \times 21.086 = 316.29$ $514 \times 429 = 220506$

Then subtract the results:

$316.29 - 220506 = -220189.71$

Step 2: Evaluate the Denominator

The denominator consists of two parts inside the square root which are then multiplied:

$\sqrt{(15 \times 26.550 - 514^2) \times (15 \times 26.191 - 429^2)}$

First, calculate each part inside the parentheses:

$15 \times 26.550 = 398.25$ $514^2 = 264196$ $15 \times 26.191 = 392.865$ $429^2 = 184041$

Now subtract these results:

$398.25 - 264196 = -263797.75$ $392.865 - 184041 = -183648.135$

Then multiply these two results:

$-263797.75 \times -183648.135 = 48460051031.91625$

Finally, take the square root of the product:

$\sqrt{48460051031.91625} = 220159.056$

Step 3: Calculate the Final Result

Divide the evaluated numerator by the evaluated denominator:

$\frac{-220189.71}{220159.056} \approx -1.000139$

Thus, the value of the given expression is approximately $-1.000139$.

Would you like more details or have any questions about this calculation? Here are five more questions you might want to ask:

1. How do you simplify complex fractions?
2. Can you explain the steps for solving quadratic equations?
3. What are the properties of logarithms?
4. How do you find the derivatives of trigonometric functions?
5. What is the process for solving systems of linear equations?

Tip: Always double-check your intermediate calculations to avoid errors in complex expressions.