Leaning against a wall is a ladder. The bottom of the ladder is 6 feet from the wall, while the top of the ladder is 10 feet above the ground. What is the ladder's length? Radical Multiplication  $$ n\sqrt{x\text{  }} $$  $$ n\sqrt{y} $$  = $$ n\sqrt{x\text{  y}} $$

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.

Leaning against a wall is a ladder. The bottom of the ladder is 6 feet from the wall, while the top of the ladder is 10 feet above the ground. What is the ladder's length?

Learn how to calculate the length of a ladder leaning against a wall using the Pythagorean Theorem. The ladder's base is 6 feet from the wall, and its top is 10 feet above the ground. Detailed steps include the Pythagorean formula and radical simplification, ideal for students in Grades 9-10.

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