This is a classic right triangle problem where the ladder forms the hypotenuse. The base of the triangle (distance from the wall) is 6 feet, and the height (vertical distance from the ground to the top of the ladder) is 10 feet.

We can use the Pythagorean Theorem to find the length of the ladder. The Pythagorean Theorem states:

$a^2 + b^2 = c^2$

Where:

• $a = 6$ feet (distance from the wall),
• $b = 10$ feet (vertical height),
• $c$ is the length of the ladder (hypotenuse).

Plugging in the values:

$6^2 + 10^2 = c^2$

$36 + 100 = c^2$

$136 = c^2$

Taking the square root of both sides:

$c = \sqrt{136} = \sqrt{4 \times 34} = 2\sqrt{34}$

Thus, the length of the ladder is $2\sqrt{34}$ feet.

Radical Multiplication Concept

You also mentioned radical multiplication, where: $n\sqrt{x} \times n\sqrt{y} = n\sqrt{xy}$

In this problem, if we express $\sqrt{136}$ using the radical multiplication rule, we can break it down as: $2\sqrt{34}$

This shows how the length of the ladder is simplified.

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Do you want further details or explanations on any step?

Here are 5 related questions to expand your understanding:

1. What if the distance from the wall was 8 feet instead of 6 feet?
2. Can the ladder's length be a rational number in this setup?
3. How does changing the height of the ladder affect its length?
4. What are some other real-world applications of the Pythagorean Theorem?
5. How would you simplify $\sqrt{200}$ using radical multiplication?

Tip: In geometry, breaking down square roots into simpler radicals often makes calculations more manageable and precise.