Solve frac{(20r^2s^3t^4)(2r^2{s}^2t)}{(-10rst)(r{s}^4{t}^2)}

$$\frac{ (20 { r }^{ 2 } { s }^{ 3 } { t }^{ 4 } )(2 { r }^{ 2 } { s }^{ 2 } t) }{ (-10rst)(r { s }^{ 4 } { t }^{ 2 } ) }$$

Evaluate

$$-4\left(rt\right)^{2}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$\frac{20r^{4}s^{3}t^{4}\times 2s^{2}t}{-10rstrs^{4}t^{2}}$$

To multiply powers of the same base, add their exponents. Add 3 and 2 to get 5.

$$\frac{20r^{4}s^{5}t^{4}\times 2t}{-10rstrs^{4}t^{2}}$$

To multiply powers of the same base, add their exponents. Add 4 and 1 to get 5.

$$\frac{20r^{4}s^{5}t^{5}\times 2}{-10rstrs^{4}t^{2}}$$

Multiply r and r to get r^{2}.

$$\frac{20r^{4}s^{5}t^{5}\times 2}{-10r^{2}sts^{4}t^{2}}$$

To multiply powers of the same base, add their exponents. Add 1 and 4 to get 5.

$$\frac{20r^{4}s^{5}t^{5}\times 2}{-10r^{2}s^{5}tt^{2}}$$

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$\frac{20r^{4}s^{5}t^{5}\times 2}{-10r^{2}s^{5}t^{3}}$$

Cancel out 10r^{2}t^{3}s^{5} in both numerator and denominator.

$$\frac{2\times 2r^{2}t^{2}}{-1}$$

Multiply 2 and 2 to get 4.

$$\frac{4r^{2}t^{2}}{-1}$$

Anything divided by -1 gives its opposite.

$$-4r^{2}t^{2}$$

Differentiate w.r.t. t

$$-8tr^{2}$$