3.2.5 \( 2 x^{2} \times x^{4}-\frac{\left(3 x^{4}\right)^{2}}{12(x)^{2}}+5 x^{6} \) \( 3.2 .6 \frac{\left(2 x y^{-2}\right)^{2}}{-3 x^{6} y^{6}} \div \frac{4 x^{2} y^{6} z^{0}}{\left(x^{2} y^{3}\right)^{4}} \)

Simplify the expression by following steps: - step0: Solution: \(2x^{2}\times x^{4}-\frac{\left(3x^{4}\right)^{2}}{12x^{2}}+5x^{6}\) - step1: Divide the terms: \(2x^{2}\times x^{4}-\frac{3^{2}x^{6}}{12}+5x^{6}\) - step2: Multiply: \(2x^{6}-\frac{3^{2}x^{6}}{12}+5x^{6}\) - step3: Evaluate the power: \(2x^{6}-\frac{9x^{6}}{12}+5x^{6}\) - step4: Add the terms: \(7x^{6}-\frac{9x^{6}}{12}\) - step5: Reduce fractions to a common denominator: \(\frac{7x^{6}\times 12}{12}-\frac{9x^{6}}{12}\) - step6: Transform the expression: \(\frac{7x^{6}\times 12-9x^{6}}{12}\) - step7: Multiply the terms: \(\frac{84x^{6}-9x^{6}}{12}\) - step8: Subtract the terms: \(\frac{75x^{6}}{12}\) - step9: Reduce the fraction: \(\frac{25x^{6}}{4}\) Calculate or simplify the expression \( (2*x*y^(-2))^2/(-3*x^6*y^6)/(4*x^2*y^6/(x^2*y^3)^4) \). Simplify the expression by following steps: - step0: Solution: \(\frac{\frac{\left(2xy^{-2}\right)^{2}}{\left(-3x^{6}y^{6}\right)}}{\left(\frac{4x^{2}y^{6}}{\left(x^{2}y^{3}\right)^{4}}\right)}\) - step1: Remove the parentheses: \(\frac{\frac{\left(2xy^{-2}\right)^{2}}{-3x^{6}y^{6}}}{\frac{4x^{2}y^{6}}{\left(x^{2}y^{3}\right)^{4}}}\) - step2: Rewrite the fraction: \(\frac{-\frac{\left(2xy^{-2}\right)^{2}}{3x^{6}y^{6}}}{\frac{4x^{2}y^{6}}{\left(x^{2}y^{3}\right)^{4}}}\) - step3: Divide the terms: \(\frac{-\frac{\left(2xy^{-2}\right)^{2}}{3x^{6}y^{6}}}{\frac{4}{x^{6}y^{6}}}\) - step4: Multiply by the reciprocal: \(-\frac{\left(2xy^{-2}\right)^{2}}{3x^{6}y^{6}}\times \frac{x^{6}y^{6}}{4}\) - step5: Rewrite the expression: \(-\frac{4x^{2}y^{-4}}{3x^{6}y^{6}}\times \frac{x^{6}y^{6}}{4}\) - step6: Reduce the fraction: \(-\frac{x^{2}y^{-4}}{3}\times 1\) - step7: Multiply the terms: \(-\frac{x^{2}y^{-4}}{3}\) - step8: Simplify: \(-\frac{x^{2}}{3y^{4}}\) Let's break down the solutions for both problems step by step. ### Problem 3.2.5 We need to simplify the expression: \[ 2 x^{2} \times x^{4} - \frac{(3 x^{4})^{2}}{12(x)^{2}} + 5 x^{6} \] 1. **First term**: \[ 2 x^{2} \times x^{4} = 2 x^{6} \] 2. **Second term**: \[ \frac{(3 x^{4})^{2}}{12(x)^{2}} = \frac{9 x^{8}}{12 x^{2}} = \frac{9 x^{6}}{12} = \frac{3 x^{6}}{4} \] 3. **Third term**: \[ 5 x^{6} \] Now, combine all the terms: \[ 2 x^{6} - \frac{3 x^{6}}{4} + 5 x^{6} \] To combine these, we can express \(2 x^{6}\) and \(5 x^{6}\) with a common denominator of 4: \[ 2 x^{6} = \frac{8 x^{6}}{4}, \quad 5 x^{6} = \frac{20 x^{6}}{4} \] Now, combine: \[ \frac{8 x^{6}}{4} - \frac{3 x^{6}}{4} + \frac{20 x^{6}}{4} = \frac{(8 - 3 + 20)x^{6}}{4} = \frac{25 x^{6}}{4} \] ### Final result for 3.2.5: \[ \frac{25 x^{6}}{4} \] --- ### Problem 3.2.6 We need to simplify the expression: \[ \frac{(2 x y^{-2})^{2}}{-3 x^{6} y^{6}} \div \frac{4 x^{2} y^{6} z^{0}}{(x^{2} y^{3})^{4}} \] 1. **Numerator of the first fraction**: \[ (2 x y^{-2})^{2} = 4 x^{2} y^{-4} \] 2. **Denominator of the first fraction**: \[ -3 x^{6} y^{6} \] So the first fraction becomes: \[ \frac{4 x^{2} y^{-4}}{-3 x^{6} y^{6}} = \frac{4 x^{2}}{-3 x^{6} y^{10}} = -\frac{4}{3} \cdot \frac{1}{x^{4} y^{10}} \] 3. **Denominator of the second fraction**: \[ (x^{2} y^{3})^{4} = x^{8} y^{12} \] Thus, the second fraction becomes: \[ \frac{4 x^{2} y^{6}}{x^{8} y^{12}} = \frac{4}{x^{6} y^{6}} \] Now, we can rewrite the division as multiplication by the reciprocal: \[ -\frac{4}{3 x^{4} y^{10}} \cdot \frac{x^{6} y^{6}}{4} \] 4. **Simplifying**: \[ -\frac{4 x^{6} y^{6}}{3 x^{4} y^{10} \cdot 4} = -\frac{x^{6}}{3 x^{4} y^{4}} = -\frac{x^{2}}{3 y^{4}} \] ### Final result for 3.2.6: \[ -\frac{x^{2}}{3 y^{4}} \] --- ### Summary of Results: - **3.2.5**: \(\frac{25 x^{6}}{4}\) - **3.2.6**: \(-\frac{x^{2}}{3 y^{4}}\)