(d) (8c^(3))/(6(c+d))÷(2c^(2))/(3c+3d)

Identify & Rewrite Division: Identify the given expression and rewrite the division as multiplication by the reciprocal of the second fraction. (8c^(3))/(6(c+d)) ÷ (2c^(2))/(3c+3d) = (8c^(3))/(6(c+d)) * (3c+3d)/(2c^(2))Simplify Coefficients & Variables: Simplify the coefficients and variables where possible. First, notice that 8 and 6 can be simplified by dividing both by 2, and 3c+3d can be factored to 3(c+d). (8c^(3))/(6(c+d)) * (3c+3d)/(2c^(2)) = (4c^(3))/(3(c+d)) * (3(c+d))/(c^(2))Cancel Common Terms: Cancel out the common terms in the numerator and the denominator. The (c+d) terms cancel out, and we are left with: (4c^(3))/(3(c+d)) * (3(c+d))/(c^(2)) = (4c^(3))/(1) * (1)/(c^(2)) = 4c^(3)/c^(2)Simplify Remaining Expression: Simplify the remaining expression by subtracting the exponents of c. When dividing like bases, subtract the exponents: c^(3)/c^(2) = c^(3-2) = c^(1) = c. 4c^(3)/c^(2) = 4c

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Q. (d)

\frac{8 c^{3}}{6(c+d)} \div \frac{2 c^{2}}{3 c+3 d}

Identify & Rewrite Division: Identify the given expression and rewrite the division as multiplication by the reciprocal of the second fraction. $\newline$ $(8c^{3})/(6(c+d)) \div (2c^{2})/(3c+3d) = (8c^{3})/(6(c+d)) \times (3c+3d)/(2c^{2})$
Simplify Coefficients & Variables: Simplify the coefficients and variables where possible. $\newline$ First, notice that $8$ and $6$ can be simplified by dividing both by $2$ , and $3c+3d$ can be factored to $3(c+d)$ . $\newline$ $\frac{8c^{3}}{6(c+d)} \times \frac{3c+3d}{2c^{2}} = \frac{4c^{3}}{3(c+d)} \times \frac{3(c+d)}{c^{2}}$
Cancel Common Terms: Cancel out the common terms in the numerator and the denominator. $\newline$ The $(c+d)$ terms cancel out, and we are left with: $\newline$ $\frac{4c^{3}}{3(c+d)} \times \frac{3(c+d)}{c^{2}} = \frac{4c^{3}}{1} \times \frac{1}{c^{2}} = \frac{4c^{3}}{c^{2}}$
Simplify Remaining Expression: Simplify the remaining expression by subtracting the exponents of $c$ . When dividing like bases, subtract the exponents: $\frac{c^{3}}{c^{2}} = c^{3-2} = c^{1} = c$ . $\frac{4c^{3}}{c^{2}} = 4c$