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

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

Evaluate

$$\frac{4s^{7}r^{9}}{9t^{14}}$$

Solution Steps

To raise \frac{2s^{2}r^{-3}}{3t^{-2}} to a power, raise both numerator and denominator to the power and then divide.

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

To raise \frac{2r^{2}}{3s^{-3}t^{4}} to a power, raise both numerator and denominator to the power and then divide.

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

Multiply \frac{\left(2s^{2}r^{-3}\right)^{-1}}{\left(3t^{-2}\right)^{-1}} times \frac{\left(2r^{2}\right)^{3}}{\left(3s^{-3}t^{4}\right)^{3}} by multiplying numerator times numerator and denominator times denominator.

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

Expand \left(2s^{2}r^{-3}\right)^{-1}.

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

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

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

To raise a power to another power, multiply the exponents. Multiply -3 and -1 to get 3.

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

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

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

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

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

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

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

Calculate 2 to the power of 3 and get 8.

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

Multiply \frac{1}{2} and 8 to get 4.

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

To multiply powers of the same base, add their exponents. Add 3 and 6 to get 9.

$$\frac{4s^{-2}r^{9}}{\left(3t^{-2}\right)^{-1}\times \left(3s^{-3}t^{4}\right)^{3}}$$

Expand \left(3t^{-2}\right)^{-1}.

$$\frac{4s^{-2}r^{9}}{3^{-1}\left(t^{-2}\right)^{-1}\times \left(3s^{-3}t^{4}\right)^{3}}$$

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

$$\frac{4s^{-2}r^{9}}{3^{-1}t^{2}\times \left(3s^{-3}t^{4}\right)^{3}}$$

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

$$\frac{4s^{-2}r^{9}}{\frac{1}{3}t^{2}\times \left(3s^{-3}t^{4}\right)^{3}}$$

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

$$\frac{4s^{-2}r^{9}}{\frac{1}{3}t^{2}\times 3^{3}\left(s^{-3}\right)^{3}\left(t^{4}\right)^{3}}$$

To raise a power to another power, multiply the exponents. Multiply -3 and 3 to get -9.

$$\frac{4s^{-2}r^{9}}{\frac{1}{3}t^{2}\times 3^{3}s^{-9}\left(t^{4}\right)^{3}}$$

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

$$\frac{4s^{-2}r^{9}}{\frac{1}{3}t^{2}\times 3^{3}s^{-9}t^{12}}$$

Calculate 3 to the power of 3 and get 27.

$$\frac{4s^{-2}r^{9}}{\frac{1}{3}t^{2}\times 27s^{-9}t^{12}}$$

Multiply \frac{1}{3} and 27 to get 9.

$$\frac{4s^{-2}r^{9}}{9t^{2}s^{-9}t^{12}}$$

To multiply powers of the same base, add their exponents. Add 2 and 12 to get 14.

$$\frac{4s^{-2}r^{9}}{9t^{14}s^{-9}}$$

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

$$\frac{4s^{7}r^{9}}{9t^{14}}$$

Expand

$$\frac{4s^{7}r^{9}}{9t^{14}}$$

Solution Steps

To raise \frac{2s^{2}r^{-3}}{3t^{-2}} to a power, raise both numerator and denominator to the power and then divide.

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

To raise \frac{2r^{2}}{3s^{-3}t^{4}} to a power, raise both numerator and denominator to the power and then divide.

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

Multiply \frac{\left(2s^{2}r^{-3}\right)^{-1}}{\left(3t^{-2}\right)^{-1}} times \frac{\left(2r^{2}\right)^{3}}{\left(3s^{-3}t^{4}\right)^{3}} by multiplying numerator times numerator and denominator times denominator.

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

Expand \left(2s^{2}r^{-3}\right)^{-1}.

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

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

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

To raise a power to another power, multiply the exponents. Multiply -3 and -1 to get 3.

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

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

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

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

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

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

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

Calculate 2 to the power of 3 and get 8.

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

Multiply \frac{1}{2} and 8 to get 4.

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

To multiply powers of the same base, add their exponents. Add 3 and 6 to get 9.

$$\frac{4s^{-2}r^{9}}{\left(3t^{-2}\right)^{-1}\times \left(3s^{-3}t^{4}\right)^{3}}$$

Expand \left(3t^{-2}\right)^{-1}.

$$\frac{4s^{-2}r^{9}}{3^{-1}\left(t^{-2}\right)^{-1}\times \left(3s^{-3}t^{4}\right)^{3}}$$

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

$$\frac{4s^{-2}r^{9}}{3^{-1}t^{2}\times \left(3s^{-3}t^{4}\right)^{3}}$$

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

$$\frac{4s^{-2}r^{9}}{\frac{1}{3}t^{2}\times \left(3s^{-3}t^{4}\right)^{3}}$$

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

$$\frac{4s^{-2}r^{9}}{\frac{1}{3}t^{2}\times 3^{3}\left(s^{-3}\right)^{3}\left(t^{4}\right)^{3}}$$

To raise a power to another power, multiply the exponents. Multiply -3 and 3 to get -9.

$$\frac{4s^{-2}r^{9}}{\frac{1}{3}t^{2}\times 3^{3}s^{-9}\left(t^{4}\right)^{3}}$$

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

$$\frac{4s^{-2}r^{9}}{\frac{1}{3}t^{2}\times 3^{3}s^{-9}t^{12}}$$

Calculate 3 to the power of 3 and get 27.

$$\frac{4s^{-2}r^{9}}{\frac{1}{3}t^{2}\times 27s^{-9}t^{12}}$$

Multiply \frac{1}{3} and 27 to get 9.

$$\frac{4s^{-2}r^{9}}{9t^{2}s^{-9}t^{12}}$$

To multiply powers of the same base, add their exponents. Add 2 and 12 to get 14.

$$\frac{4s^{-2}r^{9}}{9t^{14}s^{-9}}$$

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

$$\frac{4s^{7}r^{9}}{9t^{14}}$$