To find the hypotenuse $c$ of a right triangle with legs $a = 18$ and $b = 27$, we use the Pythagorean Theorem:

$c = \sqrt{a^2 + b^2}$

Step-by-Step Solution

1. Substitute the values of $a$ and $b$: $c = \sqrt{18^2 + 27^2}$

2. Calculate $18^2$ and $27^2$: $18^2 = 324$ $27^2 = 729$

3. Add these results: $c = \sqrt{324 + 729} = \sqrt{1053}$

4. Simplify $\sqrt{1053}$ if possible.

The prime factorization of $1053$ is $3^2 \times 3 \times 3 \times 13 = 9 \times 13 \times 9$, which doesn’t yield a perfect square simplification.

Thus, the exact answer for the hypotenuse $c$ is: $c = \sqrt{1053}$

Would you like further details on simplifying radicals or any other part of this solution?

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Here are some related questions to explore:

1. What is the hypotenuse if both legs are multiples of a smaller right triangle?
2. How does changing one leg affect the hypotenuse in a right triangle?
3. Can you use the Pythagorean theorem to solve for missing leg values?
4. How do you verify if three given sides form a right triangle?
5. What is the relationship between the area of a right triangle and its legs?

Tip: For simplifying radicals, look for factors that are perfect squares, as they can be taken outside the radical.