Solve frac{(4s^{3}t^{2})^{2}}{8(s^{4}t)^{3}}divfrac{16s^{-1}t^3}{(2s^-3t^-4)^-2}

$$\frac { ( 4 s ^ { 3 } t ^ { 2 } ) ^ { 2 } } { 8 ( s ^ { 4 } t ) ^ { 3 } } \div \frac { 16 s ^ { - 1 } t ^ { 3 } } { ( 2 s ^ { - 3 } t ^ { - 4 } ) ^ { - 2 } }$$

Evaluate

$$\frac{st^{6}}{32}$$

Solution Steps

Divide \frac{\left(4s^{3}t^{2}\right)^{2}}{8\left(s^{4}t\right)^{3}} by \frac{16s^{-1}t^{3}}{\left(2s^{-3}t^{-4}\right)^{-2}} by multiplying \frac{\left(4s^{3}t^{2}\right)^{2}}{8\left(s^{4}t\right)^{3}} by the reciprocal of \frac{16s^{-1}t^{3}}{\left(2s^{-3}t^{-4}\right)^{-2}}.

$$\frac{\left(4s^{3}t^{2}\right)^{2}\times \left(2s^{-3}t^{-4}\right)^{-2}}{8\left(s^{4}t\right)^{3}\times 16s^{-1}t^{3}}$$

Expand \left(4s^{3}t^{2}\right)^{2}.

$$\frac{4^{2}\left(s^{3}\right)^{2}\left(t^{2}\right)^{2}\times \left(2s^{-3}t^{-4}\right)^{-2}}{8\left(s^{4}t\right)^{3}\times 16s^{-1}t^{3}}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$\frac{4^{2}s^{6}\left(t^{2}\right)^{2}\times \left(2s^{-3}t^{-4}\right)^{-2}}{8\left(s^{4}t\right)^{3}\times 16s^{-1}t^{3}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\frac{4^{2}s^{6}t^{4}\times \left(2s^{-3}t^{-4}\right)^{-2}}{8\left(s^{4}t\right)^{3}\times 16s^{-1}t^{3}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{16s^{6}t^{4}\times \left(2s^{-3}t^{-4}\right)^{-2}}{8\left(s^{4}t\right)^{3}\times 16s^{-1}t^{3}}$$

Expand \left(2s^{-3}t^{-4}\right)^{-2}.

$$\frac{16s^{6}t^{4}\times 2^{-2}\left(s^{-3}\right)^{-2}\left(t^{-4}\right)^{-2}}{8\left(s^{4}t\right)^{3}\times 16s^{-1}t^{3}}$$

To raise a power to another power, multiply the exponents. Multiply -3 and -2 to get 6.

$$\frac{16s^{6}t^{4}\times 2^{-2}s^{6}\left(t^{-4}\right)^{-2}}{8\left(s^{4}t\right)^{3}\times 16s^{-1}t^{3}}$$

To raise a power to another power, multiply the exponents. Multiply -4 and -2 to get 8.

$$\frac{16s^{6}t^{4}\times 2^{-2}s^{6}t^{8}}{8\left(s^{4}t\right)^{3}\times 16s^{-1}t^{3}}$$

Calculate 2 to the power of -2 and get \frac{1}{4}.

$$\frac{16s^{6}t^{4}\times \frac{1}{4}s^{6}t^{8}}{8\left(s^{4}t\right)^{3}\times 16s^{-1}t^{3}}$$

Multiply 16 and \frac{1}{4} to get 4.

$$\frac{4s^{6}t^{4}s^{6}t^{8}}{8\left(s^{4}t\right)^{3}\times 16s^{-1}t^{3}}$$

To multiply powers of the same base, add their exponents. Add 6 and 6 to get 12.

$$\frac{4s^{12}t^{4}t^{8}}{8\left(s^{4}t\right)^{3}\times 16s^{-1}t^{3}}$$

To multiply powers of the same base, add their exponents. Add 4 and 8 to get 12.

$$\frac{4s^{12}t^{12}}{8\left(s^{4}t\right)^{3}\times 16s^{-1}t^{3}}$$

Expand \left(s^{4}t\right)^{3}.

$$\frac{4s^{12}t^{12}}{8\left(s^{4}\right)^{3}t^{3}\times 16s^{-1}t^{3}}$$

To raise a power to another power, multiply the exponents. Multiply 4 and 3 to get 12.

$$\frac{4s^{12}t^{12}}{8s^{12}t^{3}\times 16s^{-1}t^{3}}$$

Multiply 8 and 16 to get 128.

$$\frac{4s^{12}t^{12}}{128s^{12}t^{3}s^{-1}t^{3}}$$

To multiply powers of the same base, add their exponents. Add 12 and -1 to get 11.

$$\frac{4s^{12}t^{12}}{128s^{11}t^{3}t^{3}}$$

To multiply powers of the same base, add their exponents. Add 3 and 3 to get 6.

$$\frac{4s^{12}t^{12}}{128s^{11}t^{6}}$$

Cancel out 4t^{6}s^{11} in both numerator and denominator.

$$\frac{st^{6}}{32}$$

Expand

$$\frac{st^{6}}{32}$$

Solution Steps

Divide \frac{\left(4s^{3}t^{2}\right)^{2}}{8\left(s^{4}t\right)^{3}} by \frac{16s^{-1}t^{3}}{\left(2s^{-3}t^{-4}\right)^{-2}} by multiplying \frac{\left(4s^{3}t^{2}\right)^{2}}{8\left(s^{4}t\right)^{3}} by the reciprocal of \frac{16s^{-1}t^{3}}{\left(2s^{-3}t^{-4}\right)^{-2}}.

$$\frac{\left(4s^{3}t^{2}\right)^{2}\times \left(2s^{-3}t^{-4}\right)^{-2}}{8\left(s^{4}t\right)^{3}\times 16s^{-1}t^{3}}$$

Expand \left(4s^{3}t^{2}\right)^{2}.

$$\frac{4^{2}\left(s^{3}\right)^{2}\left(t^{2}\right)^{2}\times \left(2s^{-3}t^{-4}\right)^{-2}}{8\left(s^{4}t\right)^{3}\times 16s^{-1}t^{3}}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$\frac{4^{2}s^{6}\left(t^{2}\right)^{2}\times \left(2s^{-3}t^{-4}\right)^{-2}}{8\left(s^{4}t\right)^{3}\times 16s^{-1}t^{3}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\frac{4^{2}s^{6}t^{4}\times \left(2s^{-3}t^{-4}\right)^{-2}}{8\left(s^{4}t\right)^{3}\times 16s^{-1}t^{3}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{16s^{6}t^{4}\times \left(2s^{-3}t^{-4}\right)^{-2}}{8\left(s^{4}t\right)^{3}\times 16s^{-1}t^{3}}$$

Expand \left(2s^{-3}t^{-4}\right)^{-2}.

$$\frac{16s^{6}t^{4}\times 2^{-2}\left(s^{-3}\right)^{-2}\left(t^{-4}\right)^{-2}}{8\left(s^{4}t\right)^{3}\times 16s^{-1}t^{3}}$$

To raise a power to another power, multiply the exponents. Multiply -3 and -2 to get 6.

$$\frac{16s^{6}t^{4}\times 2^{-2}s^{6}\left(t^{-4}\right)^{-2}}{8\left(s^{4}t\right)^{3}\times 16s^{-1}t^{3}}$$

To raise a power to another power, multiply the exponents. Multiply -4 and -2 to get 8.

$$\frac{16s^{6}t^{4}\times 2^{-2}s^{6}t^{8}}{8\left(s^{4}t\right)^{3}\times 16s^{-1}t^{3}}$$

Calculate 2 to the power of -2 and get \frac{1}{4}.

$$\frac{16s^{6}t^{4}\times \frac{1}{4}s^{6}t^{8}}{8\left(s^{4}t\right)^{3}\times 16s^{-1}t^{3}}$$

Multiply 16 and \frac{1}{4} to get 4.

$$\frac{4s^{6}t^{4}s^{6}t^{8}}{8\left(s^{4}t\right)^{3}\times 16s^{-1}t^{3}}$$

To multiply powers of the same base, add their exponents. Add 6 and 6 to get 12.

$$\frac{4s^{12}t^{4}t^{8}}{8\left(s^{4}t\right)^{3}\times 16s^{-1}t^{3}}$$

To multiply powers of the same base, add their exponents. Add 4 and 8 to get 12.

$$\frac{4s^{12}t^{12}}{8\left(s^{4}t\right)^{3}\times 16s^{-1}t^{3}}$$

Expand \left(s^{4}t\right)^{3}.

$$\frac{4s^{12}t^{12}}{8\left(s^{4}\right)^{3}t^{3}\times 16s^{-1}t^{3}}$$

To raise a power to another power, multiply the exponents. Multiply 4 and 3 to get 12.

$$\frac{4s^{12}t^{12}}{8s^{12}t^{3}\times 16s^{-1}t^{3}}$$

Multiply 8 and 16 to get 128.

$$\frac{4s^{12}t^{12}}{128s^{12}t^{3}s^{-1}t^{3}}$$

To multiply powers of the same base, add their exponents. Add 12 and -1 to get 11.

$$\frac{4s^{12}t^{12}}{128s^{11}t^{3}t^{3}}$$

To multiply powers of the same base, add their exponents. Add 3 and 3 to get 6.

$$\frac{4s^{12}t^{12}}{128s^{11}t^{6}}$$

Cancel out 4t^{6}s^{11} in both numerator and denominator.

$$\frac{st^{6}}{32}$$