Solve [frac{(2m^{2}divn^{3}p)^3}{m^6n^8}]^-2

$$[ \frac { ( 2 m ^ { 2 } \div n ^ { 3 } p ) ^ { 3 } } { m ^ { 6 } n ^ { 8 } } ] ^ { - 2 }$$

Evaluate

$$\frac{n^{34}}{64p^{6}}$$

Solution Steps

Express \frac{2m^{2}}{n^{3}}p as a single fraction.

$$\left(\frac{\left(\frac{2m^{2}p}{n^{3}}\right)^{3}}{m^{6}n^{8}}\right)^{-2}$$

To raise \frac{2m^{2}p}{n^{3}} to a power, raise both numerator and denominator to the power and then divide.

$$\left(\frac{\frac{\left(2m^{2}p\right)^{3}}{\left(n^{3}\right)^{3}}}{m^{6}n^{8}}\right)^{-2}$$

Express \frac{\frac{\left(2m^{2}p\right)^{3}}{\left(n^{3}\right)^{3}}}{m^{6}n^{8}} as a single fraction.

$$\left(\frac{\left(2m^{2}p\right)^{3}}{\left(n^{3}\right)^{3}m^{6}n^{8}}\right)^{-2}$$

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

$$\left(\frac{2^{3}\left(m^{2}\right)^{3}p^{3}}{\left(n^{3}\right)^{3}m^{6}n^{8}}\right)^{-2}$$

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

$$\left(\frac{2^{3}m^{6}p^{3}}{\left(n^{3}\right)^{3}m^{6}n^{8}}\right)^{-2}$$

Calculate 2 to the power of 3 and get 8.

$$\left(\frac{8m^{6}p^{3}}{\left(n^{3}\right)^{3}m^{6}n^{8}}\right)^{-2}$$

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

$$\left(\frac{8m^{6}p^{3}}{n^{9}m^{6}n^{8}}\right)^{-2}$$

To multiply powers of the same base, add their exponents. Add 9 and 8 to get 17.

$$\left(\frac{8m^{6}p^{3}}{n^{17}m^{6}}\right)^{-2}$$

Cancel out m^{6} in both numerator and denominator.

$$\left(\frac{8p^{3}}{n^{17}}\right)^{-2}$$

To raise \frac{8p^{3}}{n^{17}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(8p^{3}\right)^{-2}}{\left(n^{17}\right)^{-2}}$$

To raise a power to another power, multiply the exponents. Multiply 17 and -2 to get -34.

$$\frac{\left(8p^{3}\right)^{-2}}{n^{-34}}$$

Expand \left(8p^{3}\right)^{-2}.

$$\frac{8^{-2}\left(p^{3}\right)^{-2}}{n^{-34}}$$

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

$$\frac{8^{-2}p^{-6}}{n^{-34}}$$

Calculate 8 to the power of -2 and get \frac{1}{64}.

$$\frac{\frac{1}{64}p^{-6}}{n^{-34}}$$

Expand

$$\frac{n^{34}}{64p^{6}}$$

Solution Steps

Express \frac{2m^{2}}{n^{3}}p as a single fraction.

$$\left(\frac{\left(\frac{2m^{2}p}{n^{3}}\right)^{3}}{m^{6}n^{8}}\right)^{-2}$$

To raise \frac{2m^{2}p}{n^{3}} to a power, raise both numerator and denominator to the power and then divide.

$$\left(\frac{\frac{\left(2m^{2}p\right)^{3}}{\left(n^{3}\right)^{3}}}{m^{6}n^{8}}\right)^{-2}$$

Express \frac{\frac{\left(2m^{2}p\right)^{3}}{\left(n^{3}\right)^{3}}}{m^{6}n^{8}} as a single fraction.

$$\left(\frac{\left(2m^{2}p\right)^{3}}{\left(n^{3}\right)^{3}m^{6}n^{8}}\right)^{-2}$$

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

$$\left(\frac{2^{3}\left(m^{2}\right)^{3}p^{3}}{\left(n^{3}\right)^{3}m^{6}n^{8}}\right)^{-2}$$

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

$$\left(\frac{2^{3}m^{6}p^{3}}{\left(n^{3}\right)^{3}m^{6}n^{8}}\right)^{-2}$$

Calculate 2 to the power of 3 and get 8.

$$\left(\frac{8m^{6}p^{3}}{\left(n^{3}\right)^{3}m^{6}n^{8}}\right)^{-2}$$

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

$$\left(\frac{8m^{6}p^{3}}{n^{9}m^{6}n^{8}}\right)^{-2}$$

To multiply powers of the same base, add their exponents. Add 9 and 8 to get 17.

$$\left(\frac{8m^{6}p^{3}}{n^{17}m^{6}}\right)^{-2}$$

Cancel out m^{6} in both numerator and denominator.

$$\left(\frac{8p^{3}}{n^{17}}\right)^{-2}$$

To raise \frac{8p^{3}}{n^{17}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(8p^{3}\right)^{-2}}{\left(n^{17}\right)^{-2}}$$

To raise a power to another power, multiply the exponents. Multiply 17 and -2 to get -34.

$$\frac{\left(8p^{3}\right)^{-2}}{n^{-34}}$$

Expand \left(8p^{3}\right)^{-2}.

$$\frac{8^{-2}\left(p^{3}\right)^{-2}}{n^{-34}}$$

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

$$\frac{8^{-2}p^{-6}}{n^{-34}}$$

Calculate 8 to the power of -2 and get \frac{1}{64}.

$$\frac{\frac{1}{64}p^{-6}}{n^{-34}}$$