1. The problem is to verify or simplify the expression $2\sqrt{13} \times 3\sqrt{12} \times 4\sqrt{12}$ and compare it to $3^2 \times 4^3$. 2. First, recall that $a\sqrt{b} = a \times b^{1/2}$, so rewrite each term: $$2\sqrt{13} = 2 \times 13^{1/2}$$ $$3\sqrt{12} = 3 \times 12^{1/2}$$ $$4\sqrt{12} = 4 \times 12^{1/2}$$ 3. Multiply all terms together: $$2 \times 13^{1/2} \times 3 \times 12^{1/2} \times 4 \times 12^{1/2} = (2 \times 3 \times 4) \times 13^{1/2} \times 12^{1/2} \times 12^{1/2}$$ 4. Simplify the coefficients: $$2 \times 3 \times 4 = 24$$ 5. Combine the radicals with the same base: $$12^{1/2} \times 12^{1/2} = 12^{1/2 + 1/2} = 12^1 = 12$$ 6. So the expression becomes: $$24 \times 13^{1/2} \times 12$$ 7. Multiply the constants: $$24 \times 12 = 288$$ 8. Final expression: $$288 \times \sqrt{13}$$ 9. Now evaluate the right side: $$3^2 \times 4^3 = 9 \times 64 = 576$$ 10. The left side is $288 \sqrt{13}$, and the right side is $576$. Since $\sqrt{13} \approx 3.605$, the left side is approximately: $$288 \times 3.605 = 1038.24$$ 11. Therefore, the two sides are not equal. Final answer: $2\sqrt{13} \times 3\sqrt{12} \times 4\sqrt{12} = 288 \sqrt{13} \neq 3^2 \times 4^3 = 576$