Solve frac{(r^{2}s^{-1})^-1(r^-3s)^2}{(4s)^-3}

$$\frac { ( r ^ { 2 } s ^ { - 1 } ) ^ { - 1 } ( r ^ { - 3 } s ) ^ { 2 } } { ( 4 s ) ^ { - 3 } }$$

Evaluate

$$\frac{64s^{6}}{r^{8}}$$

Solution Steps

Expand \left(r^{2}s^{-1}\right)^{-1}.

$$\frac{\left(r^{2}\right)^{-1}\left(s^{-1}\right)^{-1}\left(r^{-3}s\right)^{2}}{\left(4s\right)^{-3}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and -1 to get -2.

$$\frac{r^{-2}\left(s^{-1}\right)^{-1}\left(r^{-3}s\right)^{2}}{\left(4s\right)^{-3}}$$

To raise a power to another power, multiply the exponents. Multiply -1 and -1 to get 1.

$$\frac{r^{-2}s^{1}\left(r^{-3}s\right)^{2}}{\left(4s\right)^{-3}}$$

Calculate s to the power of 1 and get s.

$$\frac{r^{-2}s\left(r^{-3}s\right)^{2}}{\left(4s\right)^{-3}}$$

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

$$\frac{r^{-2}s\left(r^{-3}\right)^{2}s^{2}}{\left(4s\right)^{-3}}$$

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

$$\frac{r^{-2}sr^{-6}s^{2}}{\left(4s\right)^{-3}}$$

To multiply powers of the same base, add their exponents. Add -2 and -6 to get -8.

$$\frac{r^{-8}ss^{2}}{\left(4s\right)^{-3}}$$

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

$$\frac{r^{-8}s^{3}}{\left(4s\right)^{-3}}$$

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

$$\frac{r^{-8}s^{3}}{4^{-3}s^{-3}}$$

Calculate 4 to the power of -3 and get \frac{1}{64}.

$$\frac{r^{-8}s^{3}}{\frac{1}{64}s^{-3}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\frac{r^{-8}s^{6}}{\frac{1}{64}}$$

Divide r^{-8}s^{6} by \frac{1}{64} by multiplying r^{-8}s^{6} by the reciprocal of \frac{1}{64}.

$$r^{-8}s^{6}\times 64$$

Expand

$$\frac{64s^{6}}{r^{8}}$$

Solution Steps

Expand \left(r^{2}s^{-1}\right)^{-1}.

$$\frac{\left(r^{2}\right)^{-1}\left(s^{-1}\right)^{-1}\left(r^{-3}s\right)^{2}}{\left(4s\right)^{-3}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and -1 to get -2.

$$\frac{r^{-2}\left(s^{-1}\right)^{-1}\left(r^{-3}s\right)^{2}}{\left(4s\right)^{-3}}$$

To raise a power to another power, multiply the exponents. Multiply -1 and -1 to get 1.

$$\frac{r^{-2}s^{1}\left(r^{-3}s\right)^{2}}{\left(4s\right)^{-3}}$$

Calculate s to the power of 1 and get s.

$$\frac{r^{-2}s\left(r^{-3}s\right)^{2}}{\left(4s\right)^{-3}}$$

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

$$\frac{r^{-2}s\left(r^{-3}\right)^{2}s^{2}}{\left(4s\right)^{-3}}$$

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

$$\frac{r^{-2}sr^{-6}s^{2}}{\left(4s\right)^{-3}}$$

To multiply powers of the same base, add their exponents. Add -2 and -6 to get -8.

$$\frac{r^{-8}ss^{2}}{\left(4s\right)^{-3}}$$

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

$$\frac{r^{-8}s^{3}}{\left(4s\right)^{-3}}$$

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

$$\frac{r^{-8}s^{3}}{4^{-3}s^{-3}}$$

Calculate 4 to the power of -3 and get \frac{1}{64}.

$$\frac{r^{-8}s^{3}}{\frac{1}{64}s^{-3}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\frac{r^{-8}s^{6}}{\frac{1}{64}}$$

Divide r^{-8}s^{6} by \frac{1}{64} by multiplying r^{-8}s^{6} by the reciprocal of \frac{1}{64}.

$$r^{-8}s^{6}\times 64$$