The expression you provided is:

$\left( 15^2 - 9^2 \right)^{0.5}$

Step 1: Apply the difference of squares formula

The expression inside the parentheses, $15^2 - 9^2$, is a difference of squares. The formula for this is:

$a^2 - b^2 = (a - b)(a + b)$

So, applying this to $15^2 - 9^2$:

$15^2 - 9^2 = (15 - 9)(15 + 9) = 6 \times 24 = 144$

Step 2: Take the square root

Now, we take the square root of 144:

$\sqrt{144} = 12$

Final answer:

$\left( 15^2 - 9^2 \right)^{0.5} = 12$

Would you like further explanation on any steps or have any other questions? Here are some related questions:

1. What is the difference of squares formula and how is it derived?
2. How do you calculate square roots of non-perfect squares?
3. What happens if you have a negative value under the square root?
4. Can you simplify other algebraic expressions using the difference of squares?
5. How do you solve similar problems involving cube roots or higher powers?

Tip: The difference of squares formula is a very useful tool to simplify expressions and quickly solve problems involving quadratic terms.