6) \( 4^{4}+3\{-6+2[5-4(3-2)+5(7-8)]-\sqrt{144}\}=2 \)

Let's simplify the given expression step by step. 1. Start with the innermost parentheses: \( 3 - 2 = 1 \) Therefore, \( 4(3-2) = 4(1) = 4 \). 2. Next, calculate \( 5 - 4(3 - 2) + 5(7 - 8) \): We already computed \( 4(3-2) = 4 \). Now, compute \( 7 - 8 = -1 \), and so \( 5(7 - 8) = 5 \times -1 = -5 \). Thus: \( 5 - 4 + (-5) = 5 - 4 - 5 = -4 \). 3. Now substitute back into the expression: \( 3\{-6 + 2[-4] - \sqrt{144}\} \). Let's compute \( \sqrt{144} = 12 \). Now substitute: \( 3\{-6 + 2[-4] - 12\} \). 4. Calculate \( 2[-4] = -8 \): Thus we have \( -6 - 8 - 12 = -26 \). Substitute this back: \( 3(-26) = -78 \). 5. Finally, compute \( 4^4 - 78 \): Since \( 4^4 = 256 \), we get: \( 256 - 78 = 178 \). So the full simplification can be concluded as follows: \[ 4^{4} + 3\{-6 + 2[5 - 4(3 - 2) + 5(7 - 8)] - \sqrt{144}\} = 178 \] Now we can see that \[ 178 = 2 \] is not true. The original equation does not hold, indicating that either the equation is incorrect or misleading.