The expression in the image is:

$\left(\sqrt{9 \times 32}\right)^3$

Let’s solve this step-by-step.

Step 1: Simplify Inside the Square Root

First, calculate $9 \times 32$: $9 \times 32 = 288$ So, we have: $\sqrt{9 \times 32} = \sqrt{288}$

Step 2: Simplify $\sqrt{288}$

To simplify $\sqrt{288}$, notice that $288 = 144 \times 2 = (12)^2 \times 2$. Thus: $\sqrt{288} = \sqrt{(12)^2 \times 2} = 12 \sqrt{2}$

Step 3: Cube the Result

Now, substitute $\sqrt{288} = 12 \sqrt{2}$ back into the expression: $\left(12 \sqrt{2}\right)^3$

Expand the cube: $\left(12 \sqrt{2}\right)^3 = 12^3 \times (\sqrt{2})^3$

Calculating each part:

• $12^3 = 1728$
• $(\sqrt{2})^3 = 2^{3/2} = 2 \sqrt{2}$

So: $12^3 \times (\sqrt{2})^3 = 1728 \times 2 \sqrt{2} = 3456 \sqrt{2}$

Final Answer

$\left(\sqrt{9 \times 32}\right)^3 = 3456 \sqrt{2}$

Would you like further details or have any questions?

Additional Related Questions

1. What are other ways to simplify square roots of large numbers?
2. How would you apply this method to expressions with higher powers?
3. How is this different from evaluating expressions with nested square roots?
4. How does the property $\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$ apply here?
5. Can you cube a square root directly in terms of fractional exponents?

Tip:

When simplifying square roots, look for perfect squares as factors to reduce the expression efficiently.