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Divide. (3x2)÷(3x+2)\left(\frac{3x}{2}\right)\div\left(\frac{3}{x}+2\right)

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Q. Divide. (3x2)÷(3x+2)\left(\frac{3x}{2}\right)\div\left(\frac{3}{x}+2\right)
  1. Rewrite as division problem: Rewrite the complex fraction as a division problem.\newline(3x2)/(3x+2)(\frac{3x}{2}) / (\frac{3}{x} + 2) can be rewritten as (3x2)⋅(13x+2)(\frac{3x}{2}) \cdot (\frac{1}{\frac{3}{x} + 2}).
  2. Simplify denominator: Simplify the expression in the denominator of the second fraction.\newlineTo combine the terms in the denominator, find a common denominator, which is xx.\newline(3x+2)=3+2xx(\frac{3}{x} + 2) = \frac{3 + 2x}{x}.
  3. Rewrite with simplified denominator: Rewrite the original expression with the simplified denominator.\newlineNow the expression is (3x2)×(1(3+2xx))(\frac{3x}{2}) \times (\frac{1}{(\frac{3 + 2x}{x})}).
  4. Multiply the fractions: Multiply the two fractions.\newlineTo multiply the fractions, we take the numerator of the first fraction and multiply it by the reciprocal of the second fraction.\newline(3x2)×(x3+2x)=3x×x2×(3+2x)(\frac{3x}{2}) \times (\frac{x}{3 + 2x}) = \frac{3x \times x}{2 \times (3 + 2x)}.
  5. Simplify numerator: Simplify the multiplication in the numerator.\newlineMultiply 3x3x by xx to get 3x23x^2.\newline3x22⋅(3+2x)=3x26+4x\frac{3x^2}{2 \cdot (3 + 2x)} = \frac{3x^2}{6 + 4x}.
  6. Check for further simplification: Check if the expression can be simplified further.\newlineThe expression 3x26+4x\frac{3x^2}{6 + 4x} cannot be simplified further because the numerator and denominator do not have common factors.

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