3.2.5 \( 2 x^{2} \times x^{4}-\frac{\left(3 x^{6}\right)^{2}}{12(x)^{2}}+5 x^{6} \) 3.2.6 \( \frac{\left(2 x y^{-2}\right)^{2}}{-3 x^{6} y^{6}} \div \frac{4 x^{2} y^{6} x^{0}}{\left(x^{2} y^{3}\right)^{4}} \)

Simplify the expression by following steps: - step0: Solution: \(2x^{2}\times x^{4}-\frac{\left(3x^{6}\right)^{2}}{12x^{2}}+5x^{6}\) - step1: Divide the terms: \(2x^{2}\times x^{4}-\frac{3^{2}x^{10}}{12}+5x^{6}\) - step2: Multiply: \(2x^{6}-\frac{3^{2}x^{10}}{12}+5x^{6}\) - step3: Evaluate the power: \(2x^{6}-\frac{9x^{10}}{12}+5x^{6}\) - step4: Add the terms: \(7x^{6}-\frac{9x^{10}}{12}\) - step5: Reduce fractions to a common denominator: \(\frac{7x^{6}\times 12}{12}-\frac{9x^{10}}{12}\) - step6: Transform the expression: \(\frac{7x^{6}\times 12-9x^{10}}{12}\) - step7: Multiply the terms: \(\frac{84x^{6}-9x^{10}}{12}\) - step8: Factor: \(\frac{3\left(28x^{6}-3x^{10}\right)}{12}\) - step9: Reduce the fraction: \(\frac{28x^{6}-3x^{10}}{4}\) Calculate or simplify the expression \( (2*x*y^(-2))^2/(-3*x^6*y^6)/(4*x^2*y^6*x^0/((x^2*y^3)^4)) \). Simplify the expression by following steps: - step0: Solution: \(\frac{\frac{\left(2xy^{-2}\right)^{2}}{\left(-3x^{6}y^{6}\right)}}{\left(\frac{4x^{2}y^{6}x^{0}}{\left(\left(x^{2}y^{3}\right)^{4}\right)}\right)}\) - step1: Evaluate: \(\frac{\frac{\left(2xy^{-2}\right)^{2}}{\left(-3x^{6}y^{6}\right)}}{\left(\frac{4x^{2}y^{6}x^{0}}{\left(x^{2}y^{3}\right)^{4}}\right)}\) - step2: Remove the parentheses: \(\frac{\frac{\left(2xy^{-2}\right)^{2}}{-3x^{6}y^{6}}}{\frac{4x^{2}y^{6}x^{0}}{\left(x^{2}y^{3}\right)^{4}}}\) - step3: Evaluate the power: \(\frac{\frac{\left(2xy^{-2}\right)^{2}}{-3x^{6}y^{6}}}{\frac{4x^{2}y^{6}\times 1}{\left(x^{2}y^{3}\right)^{4}}}\) - step4: Reduce the fraction: \(\frac{\frac{\left(2xy^{-2}\right)^{2}}{-3x^{6}y^{6}}}{\frac{4}{x^{6}y^{6}}}\) - step5: Rewrite the fraction: \(\frac{-\frac{\left(2xy^{-2}\right)^{2}}{3x^{6}y^{6}}}{\frac{4}{x^{6}y^{6}}}\) - step6: Multiply by the reciprocal: \(-\frac{\left(2xy^{-2}\right)^{2}}{3x^{6}y^{6}}\times \frac{x^{6}y^{6}}{4}\) - step7: Rewrite the expression: \(-\frac{4x^{2}y^{-4}}{3x^{6}y^{6}}\times \frac{x^{6}y^{6}}{4}\) - step8: Reduce the fraction: \(-\frac{x^{2}y^{-4}}{3}\times 1\) - step9: Multiply the terms: \(-\frac{x^{2}y^{-4}}{3}\) - step10: Simplify: \(-\frac{x^{2}}{3y^{4}}\) Let's break down the solutions for each expression step by step. ### Problem 3.2.5 The expression is: \[ 2 x^{2} \times x^{4} - \frac{(3 x^{6})^{2}}{12(x)^{2}} + 5 x^{6} \] 1. **First term**: \(2 x^{2} \times x^{4} = 2 x^{6}\) 2. **Second term**: - Calculate \((3 x^{6})^{2} = 9 x^{12}\) - Then, \(\frac{9 x^{12}}{12 x^{2}} = \frac{9}{12} x^{10} = \frac{3}{4} x^{10}\) 3. **Third term**: \(5 x^{6}\) Now, combine all the terms: \[ 2 x^{6} - \frac{3}{4} x^{10} + 5 x^{6} = (2 + 5)x^{6} - \frac{3}{4} x^{10} = 7 x^{6} - \frac{3}{4} x^{10} \] Thus, the simplified form is: \[ \frac{28x^{6} - 3x^{10}}{4} \] ### Problem 3.2.6 The expression is: \[ \frac{(2 x y^{-2})^{2}}{-3 x^{6} y^{6}} \div \frac{4 x^{2} y^{6} x^{0}}{(x^{2} y^{3})^{4}} \] 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: \[ \frac{4 x^{2} y^{-4}}{-3 x^{6} y^{6}} = -\frac{4 x^{2}}{3 x^{6} y^{10}} = -\frac{4}{3 x^{4} y^{10}} \] 3. **Numerator of the second fraction**: - \(4 x^{2} y^{6} x^{0} = 4 x^{2} y^{6}\) 4. **Denominator of the second fraction**: - \((x^{2} y^{3})^{4} = x^{8} y^{12}\) So, the second fraction becomes: \[ \frac{4 x^{2} y^{6}}{x^{8} y^{12}} = \frac{4}{x^{6} y^{6}} \] Now, we need to divide the first fraction by the second: \[ -\frac{4}{3 x^{4} y^{10}} \div \frac{4}{x^{6} y^{6}} = -\frac{4}{3 x^{4} y^{10}} \times \frac{x^{6} y^{6}}{4} = -\frac{x^{6}}{3 x^{4} y^{4}} = -\frac{x^{2}}{3 y^{4}} \] Thus, the simplified form is: \[ -\frac{x^{2}}{3y^{4}} \] ### Final Results - For 3.2.5: \(\frac{28x^{6} - 3x^{10}}{4}\) - For 3.2.6: \(-\frac{x^{2}}{3y^{4}}\)