a) \( 6 x^{2}-4 x^{2}+x^{2} \) b) \( -3 x+8 x-15 x \) c) \( -x^{4}-2 x^{3}+6 x^{3} \) d) \( x-3 \div 4 x+2 \div 3 x \) e) \( 5 \div 8 x^{2}-x^{2}+x^{3} \div 3 \) f) \( 4 x-2 \div 9 x+x \div 3 \)

Simplify the expression by following steps: - step0: Solution: \(-x^{4}-2x^{3}+6x^{3}\) - step1: Add the terms: \(-x^{4}+4x^{3}\) Calculate or simplify the expression \( 4 x-2 \div 9 x+x \div 3 \). Simplify the expression by following steps: - step0: Solution: \(4x-2\div \left(9x\right)+x\div 3\) - step1: Rewrite the expression: \(4x-\frac{2}{9x}+x\div 3\) - step2: Rewrite the expression: \(4x-\frac{2}{9x}+\frac{x}{3}\) - step3: Reduce fractions to a common denominator: \(\frac{4x\times 9x}{9x}-\frac{2}{9x}+\frac{x\times 3x}{3\times 3x}\) - step4: Multiply the numbers: \(\frac{4x\times 9x}{9x}-\frac{2}{9x}+\frac{x\times 3x}{9x}\) - step5: Transform the expression: \(\frac{4x\times 9x-2+x\times 3x}{9x}\) - step6: Multiply the terms: \(\frac{36x^{2}-2+x\times 3x}{9x}\) - step7: Multiply the terms: \(\frac{36x^{2}-2+3x^{2}}{9x}\) - step8: Add the terms: \(\frac{39x^{2}-2}{9x}\) Calculate or simplify the expression \( 6 x^{2}-4 x^{2}+x^{2} \). Simplify the expression by following steps: - step0: Solution: \(6x^{2}-4x^{2}+x^{2}\) - step1: Collect like terms: \(\left(6-4+1\right)x^{2}\) - step2: Calculate: \(3x^{2}\) Calculate or simplify the expression \( x-3 \div 4 x+2 \div 3 x \). Simplify the expression by following steps: - step0: Solution: \(x-3\div \left(4x\right)+2\div \left(3x\right)\) - step1: Rewrite the expression: \(x-\frac{3}{4x}+2\div \left(3x\right)\) - step2: Rewrite the expression: \(x-\frac{3}{4x}+\frac{2}{3x}\) - step3: Reduce fractions to a common denominator: \(\frac{x\times 4\times 3x}{4\times 3x}-\frac{3\times 3}{4x\times 3}+\frac{2\times 4}{3x\times 4}\) - step4: Multiply the numbers: \(\frac{x\times 4\times 3x}{12x}-\frac{3\times 3}{4x\times 3}+\frac{2\times 4}{3x\times 4}\) - step5: Multiply the numbers: \(\frac{x\times 4\times 3x}{12x}-\frac{3\times 3}{12x}+\frac{2\times 4}{3x\times 4}\) - step6: Multiply the numbers: \(\frac{x\times 4\times 3x}{12x}-\frac{3\times 3}{12x}+\frac{2\times 4}{12x}\) - step7: Transform the expression: \(\frac{x\times 4\times 3x-3\times 3+2\times 4}{12x}\) - step8: Multiply the terms: \(\frac{12x^{2}-3\times 3+2\times 4}{12x}\) - step9: Multiply the numbers: \(\frac{12x^{2}-9+2\times 4}{12x}\) - step10: Multiply the numbers: \(\frac{12x^{2}-9+8}{12x}\) - step11: Add the numbers: \(\frac{12x^{2}-1}{12x}\) Calculate or simplify the expression \( -3 x+8 x-15 x \). Simplify the expression by following steps: - step0: Solution: \(-3x+8x-15x\) - step1: Collect like terms: \(\left(-3+8-15\right)x\) - step2: Calculate: \(-10x\) Calculate or simplify the expression \( 5 \div 8 x^{2}-x^{2}+x^{3} \div 3 \). Simplify the expression by following steps: - step0: Solution: \(5\div \left(8x^{2}\right)-x^{2}+x^{3}\div 3\) - step1: Rewrite the expression: \(\frac{5}{8x^{2}}-x^{2}+x^{3}\div 3\) - step2: Rewrite the expression: \(\frac{5}{8x^{2}}-x^{2}+\frac{x^{3}}{3}\) - step3: Reduce fractions to a common denominator: \(\frac{5\times 3}{8x^{2}\times 3}-\frac{x^{2}\times 8\times 3x^{2}}{8\times 3x^{2}}+\frac{x^{3}\times 8x^{2}}{3\times 8x^{2}}\) - step4: Multiply the numbers: \(\frac{5\times 3}{24x^{2}}-\frac{x^{2}\times 8\times 3x^{2}}{8\times 3x^{2}}+\frac{x^{3}\times 8x^{2}}{3\times 8x^{2}}\) - step5: Multiply the numbers: \(\frac{5\times 3}{24x^{2}}-\frac{x^{2}\times 8\times 3x^{2}}{24x^{2}}+\frac{x^{3}\times 8x^{2}}{3\times 8x^{2}}\) - step6: Multiply the numbers: \(\frac{5\times 3}{24x^{2}}-\frac{x^{2}\times 8\times 3x^{2}}{24x^{2}}+\frac{x^{3}\times 8x^{2}}{24x^{2}}\) - step7: Transform the expression: \(\frac{5\times 3-x^{2}\times 8\times 3x^{2}+x^{3}\times 8x^{2}}{24x^{2}}\) - step8: Multiply the numbers: \(\frac{15-x^{2}\times 8\times 3x^{2}+x^{3}\times 8x^{2}}{24x^{2}}\) - step9: Multiply the terms: \(\frac{15-24x^{4}+x^{3}\times 8x^{2}}{24x^{2}}\) - step10: Multiply the terms: \(\frac{15-24x^{4}+8x^{5}}{24x^{2}}\) Aquí están las soluciones simplificadas para cada una de las expresiones: a) \( 6 x^{2}-4 x^{2}+x^{2} \) se simplifica a: \[ 3x^{2} \] b) \( -3 x+8 x-15 x \) se simplifica a: \[ -10x \] c) \( -x^{4}-2 x^{3}+6 x^{3} \) se simplifica a: \[ -x^{4}+4x^{3} \] d) \( x-3 \div 4 x+2 \div 3 x \) se simplifica a: \[ \frac{12x^{2}-1}{12x} \] e) \( 5 \div 8 x^{2}-x^{2}+x^{3} \div 3 \) se simplifica a: \[ \frac{15-24x^{4}+8x^{5}}{24x^{2}} \] f) \( 4 x-2 \div 9 x+x \div 3 \) se simplifica a: \[ \frac{39x^{2}-2}{9x} \]