Solve frac{(-2x^{4}y^{3}z^{5})}{(3x^{2}y^{4})}(frac{(3xy^{2}z^{2})^{5}(-2x^{3}z)^{-2}}{(6x^{4}y^3z^5)^-4})^2

$$\frac { ( - 2 x ^ { 4 } y ^ { 3 } z ^ { 5 } ) } { ( 3 x ^ { 2 } y ^ { 4 } ) } ( \frac { ( 3 x y ^ { 2 } z ^ { 2 } ) ^ { 5 } ( - 2 x ^ { 3 } z ) ^ { - 2 } } { ( 6 x ^ { 4 } y ^ { 3 } z ^ { 5 } ) ^ { - 4 } } ) ^ { 2 }$$

Evaluate

$$-4132485216x^{32}y^{43}z^{61}$$

Solution Steps

Cancel out x^{2}y^{3} in both numerator and denominator.

$$\frac{-2x^{2}z^{5}}{3y}\times \left(\frac{\left(3xy^{2}z^{2}\right)^{5}\left(-2x^{3}z\right)^{-2}}{\left(6x^{4}y^{3}z^{5}\right)^{-4}}\right)^{2}$$

Expand \left(3xy^{2}z^{2}\right)^{5}.

$$\frac{-2x^{2}z^{5}}{3y}\times \left(\frac{3^{5}x^{5}\left(y^{2}\right)^{5}\left(z^{2}\right)^{5}\left(-2x^{3}z\right)^{-2}}{\left(6x^{4}y^{3}z^{5}\right)^{-4}}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 5 to get 10.

$$\frac{-2x^{2}z^{5}}{3y}\times \left(\frac{3^{5}x^{5}y^{10}\left(z^{2}\right)^{5}\left(-2x^{3}z\right)^{-2}}{\left(6x^{4}y^{3}z^{5}\right)^{-4}}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 5 to get 10.

$$\frac{-2x^{2}z^{5}}{3y}\times \left(\frac{3^{5}x^{5}y^{10}z^{10}\left(-2x^{3}z\right)^{-2}}{\left(6x^{4}y^{3}z^{5}\right)^{-4}}\right)^{2}$$

Calculate 3 to the power of 5 and get 243.

$$\frac{-2x^{2}z^{5}}{3y}\times \left(\frac{243x^{5}y^{10}z^{10}\left(-2x^{3}z\right)^{-2}}{\left(6x^{4}y^{3}z^{5}\right)^{-4}}\right)^{2}$$

Expand \left(-2x^{3}z\right)^{-2}.

$$\frac{-2x^{2}z^{5}}{3y}\times \left(\frac{243x^{5}y^{10}z^{10}\left(-2\right)^{-2}\left(x^{3}\right)^{-2}z^{-2}}{\left(6x^{4}y^{3}z^{5}\right)^{-4}}\right)^{2}$$

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

$$\frac{-2x^{2}z^{5}}{3y}\times \left(\frac{243x^{5}y^{10}z^{10}\left(-2\right)^{-2}x^{-6}z^{-2}}{\left(6x^{4}y^{3}z^{5}\right)^{-4}}\right)^{2}$$

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

$$\frac{-2x^{2}z^{5}}{3y}\times \left(\frac{243x^{5}y^{10}z^{10}\times \frac{1}{4}x^{-6}z^{-2}}{\left(6x^{4}y^{3}z^{5}\right)^{-4}}\right)^{2}$$

Multiply 243 and \frac{1}{4} to get \frac{243}{4}.

$$\frac{-2x^{2}z^{5}}{3y}\times \left(\frac{\frac{243}{4}x^{5}y^{10}z^{10}x^{-6}z^{-2}}{\left(6x^{4}y^{3}z^{5}\right)^{-4}}\right)^{2}$$

To multiply powers of the same base, add their exponents. Add 5 and -6 to get -1.

$$\frac{-2x^{2}z^{5}}{3y}\times \left(\frac{\frac{243}{4}x^{-1}y^{10}z^{10}z^{-2}}{\left(6x^{4}y^{3}z^{5}\right)^{-4}}\right)^{2}$$

To multiply powers of the same base, add their exponents. Add 10 and -2 to get 8.

$$\frac{-2x^{2}z^{5}}{3y}\times \left(\frac{\frac{243}{4}x^{-1}y^{10}z^{8}}{\left(6x^{4}y^{3}z^{5}\right)^{-4}}\right)^{2}$$

Expand \left(6x^{4}y^{3}z^{5}\right)^{-4}.

$$\frac{-2x^{2}z^{5}}{3y}\times \left(\frac{\frac{243}{4}x^{-1}y^{10}z^{8}}{6^{-4}\left(x^{4}\right)^{-4}\left(y^{3}\right)^{-4}\left(z^{5}\right)^{-4}}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 4 and -4 to get -16.

$$\frac{-2x^{2}z^{5}}{3y}\times \left(\frac{\frac{243}{4}x^{-1}y^{10}z^{8}}{6^{-4}x^{-16}\left(y^{3}\right)^{-4}\left(z^{5}\right)^{-4}}\right)^{2}$$

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

$$\frac{-2x^{2}z^{5}}{3y}\times \left(\frac{\frac{243}{4}x^{-1}y^{10}z^{8}}{6^{-4}x^{-16}y^{-12}\left(z^{5}\right)^{-4}}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 5 and -4 to get -20.

$$\frac{-2x^{2}z^{5}}{3y}\times \left(\frac{\frac{243}{4}x^{-1}y^{10}z^{8}}{6^{-4}x^{-16}y^{-12}z^{-20}}\right)^{2}$$

Calculate 6 to the power of -4 and get \frac{1}{1296}.

$$\frac{-2x^{2}z^{5}}{3y}\times \left(\frac{\frac{243}{4}x^{-1}y^{10}z^{8}}{\frac{1}{1296}x^{-16}y^{-12}z^{-20}}\right)^{2}$$

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

$$\frac{-2x^{2}z^{5}}{3y}\times \left(\frac{\frac{243}{4}x^{15}y^{22}z^{28}}{\frac{1}{1296}}\right)^{2}$$

Divide \frac{243}{4}x^{15}y^{22}z^{28} by \frac{1}{1296} by multiplying \frac{243}{4}x^{15}y^{22}z^{28} by the reciprocal of \frac{1}{1296}.

$$\frac{-2x^{2}z^{5}}{3y}\times \left(\frac{243}{4}x^{15}y^{22}z^{28}\times 1296\right)^{2}$$

Expand \left(\frac{243}{4}x^{15}y^{22}z^{28}\times 1296\right)^{2}.

$$\frac{-2x^{2}z^{5}}{3y}\times \left(\frac{243}{4}\right)^{2}\left(x^{15}\right)^{2}\left(y^{22}\right)^{2}\left(z^{28}\right)^{2}\times 1296^{2}$$

To raise a power to another power, multiply the exponents. Multiply 15 and 2 to get 30.

$$\frac{-2x^{2}z^{5}}{3y}\times \left(\frac{243}{4}\right)^{2}x^{30}\left(y^{22}\right)^{2}\left(z^{28}\right)^{2}\times 1296^{2}$$

To raise a power to another power, multiply the exponents. Multiply 22 and 2 to get 44.

$$\frac{-2x^{2}z^{5}}{3y}\times \left(\frac{243}{4}\right)^{2}x^{30}y^{44}\left(z^{28}\right)^{2}\times 1296^{2}$$

To raise a power to another power, multiply the exponents. Multiply 28 and 2 to get 56.

$$\frac{-2x^{2}z^{5}}{3y}\times \left(\frac{243}{4}\right)^{2}x^{30}y^{44}z^{56}\times 1296^{2}$$

Calculate \frac{243}{4} to the power of 2 and get \frac{59049}{16}.

$$\frac{-2x^{2}z^{5}}{3y}\times \frac{59049}{16}x^{30}y^{44}z^{56}\times 1296^{2}$$

Calculate 1296 to the power of 2 and get 1679616.

$$\frac{-2x^{2}z^{5}}{3y}\times \frac{59049}{16}x^{30}y^{44}z^{56}\times 1679616$$

Multiply \frac{59049}{16} and 1679616 to get 6198727824.

$$\frac{-2x^{2}z^{5}}{3y}\times 6198727824x^{30}y^{44}z^{56}$$

Express \frac{-2x^{2}z^{5}}{3y}\times 6198727824 as a single fraction.

$$\frac{-2x^{2}z^{5}\times 6198727824}{3y}x^{30}y^{44}z^{56}$$

Cancel out 3 in both numerator and denominator.

$$\frac{-2\times 2066242608x^{2}z^{5}}{y}x^{30}y^{44}z^{56}$$

Express \frac{-2\times 2066242608x^{2}z^{5}}{y}x^{30} as a single fraction.

$$\frac{-2\times 2066242608x^{2}z^{5}x^{30}}{y}y^{44}z^{56}$$

Express \frac{-2\times 2066242608x^{2}z^{5}x^{30}}{y}y^{44} as a single fraction.

$$\frac{-2\times 2066242608x^{2}z^{5}x^{30}y^{44}}{y}z^{56}$$

Cancel out y in both numerator and denominator.

$$-2\times 2066242608x^{2}z^{5}x^{30}y^{43}z^{56}$$

To multiply powers of the same base, add their exponents. Add 2 and 30 to get 32.

$$-2\times 2066242608x^{32}z^{5}y^{43}z^{56}$$

Multiply -2 and 2066242608 to get -4132485216.

$$-4132485216x^{32}z^{5}y^{43}z^{56}$$

To multiply powers of the same base, add their exponents. Add 5 and 56 to get 61.

$$-4132485216x^{32}z^{61}y^{43}$$

Expand

$$-4132485216x^{32}y^{43}z^{61}$$

Solution Steps

Cancel out x^{2}y^{3} in both numerator and denominator.

$$\frac{-2x^{2}z^{5}}{3y}\times \left(\frac{\left(3xy^{2}z^{2}\right)^{5}\left(-2x^{3}z\right)^{-2}}{\left(6x^{4}y^{3}z^{5}\right)^{-4}}\right)^{2}$$

Expand \left(3xy^{2}z^{2}\right)^{5}.

$$\frac{-2x^{2}z^{5}}{3y}\times \left(\frac{3^{5}x^{5}\left(y^{2}\right)^{5}\left(z^{2}\right)^{5}\left(-2x^{3}z\right)^{-2}}{\left(6x^{4}y^{3}z^{5}\right)^{-4}}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 5 to get 10.

$$\frac{-2x^{2}z^{5}}{3y}\times \left(\frac{3^{5}x^{5}y^{10}\left(z^{2}\right)^{5}\left(-2x^{3}z\right)^{-2}}{\left(6x^{4}y^{3}z^{5}\right)^{-4}}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 5 to get 10.

$$\frac{-2x^{2}z^{5}}{3y}\times \left(\frac{3^{5}x^{5}y^{10}z^{10}\left(-2x^{3}z\right)^{-2}}{\left(6x^{4}y^{3}z^{5}\right)^{-4}}\right)^{2}$$

Calculate 3 to the power of 5 and get 243.

$$\frac{-2x^{2}z^{5}}{3y}\times \left(\frac{243x^{5}y^{10}z^{10}\left(-2x^{3}z\right)^{-2}}{\left(6x^{4}y^{3}z^{5}\right)^{-4}}\right)^{2}$$

Expand \left(-2x^{3}z\right)^{-2}.

$$\frac{-2x^{2}z^{5}}{3y}\times \left(\frac{243x^{5}y^{10}z^{10}\left(-2\right)^{-2}\left(x^{3}\right)^{-2}z^{-2}}{\left(6x^{4}y^{3}z^{5}\right)^{-4}}\right)^{2}$$

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

$$\frac{-2x^{2}z^{5}}{3y}\times \left(\frac{243x^{5}y^{10}z^{10}\left(-2\right)^{-2}x^{-6}z^{-2}}{\left(6x^{4}y^{3}z^{5}\right)^{-4}}\right)^{2}$$

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

$$\frac{-2x^{2}z^{5}}{3y}\times \left(\frac{243x^{5}y^{10}z^{10}\times \frac{1}{4}x^{-6}z^{-2}}{\left(6x^{4}y^{3}z^{5}\right)^{-4}}\right)^{2}$$

Multiply 243 and \frac{1}{4} to get \frac{243}{4}.

$$\frac{-2x^{2}z^{5}}{3y}\times \left(\frac{\frac{243}{4}x^{5}y^{10}z^{10}x^{-6}z^{-2}}{\left(6x^{4}y^{3}z^{5}\right)^{-4}}\right)^{2}$$

To multiply powers of the same base, add their exponents. Add 5 and -6 to get -1.

$$\frac{-2x^{2}z^{5}}{3y}\times \left(\frac{\frac{243}{4}x^{-1}y^{10}z^{10}z^{-2}}{\left(6x^{4}y^{3}z^{5}\right)^{-4}}\right)^{2}$$

To multiply powers of the same base, add their exponents. Add 10 and -2 to get 8.

$$\frac{-2x^{2}z^{5}}{3y}\times \left(\frac{\frac{243}{4}x^{-1}y^{10}z^{8}}{\left(6x^{4}y^{3}z^{5}\right)^{-4}}\right)^{2}$$

Expand \left(6x^{4}y^{3}z^{5}\right)^{-4}.

$$\frac{-2x^{2}z^{5}}{3y}\times \left(\frac{\frac{243}{4}x^{-1}y^{10}z^{8}}{6^{-4}\left(x^{4}\right)^{-4}\left(y^{3}\right)^{-4}\left(z^{5}\right)^{-4}}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 4 and -4 to get -16.

$$\frac{-2x^{2}z^{5}}{3y}\times \left(\frac{\frac{243}{4}x^{-1}y^{10}z^{8}}{6^{-4}x^{-16}\left(y^{3}\right)^{-4}\left(z^{5}\right)^{-4}}\right)^{2}$$

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

$$\frac{-2x^{2}z^{5}}{3y}\times \left(\frac{\frac{243}{4}x^{-1}y^{10}z^{8}}{6^{-4}x^{-16}y^{-12}\left(z^{5}\right)^{-4}}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 5 and -4 to get -20.

$$\frac{-2x^{2}z^{5}}{3y}\times \left(\frac{\frac{243}{4}x^{-1}y^{10}z^{8}}{6^{-4}x^{-16}y^{-12}z^{-20}}\right)^{2}$$

Calculate 6 to the power of -4 and get \frac{1}{1296}.

$$\frac{-2x^{2}z^{5}}{3y}\times \left(\frac{\frac{243}{4}x^{-1}y^{10}z^{8}}{\frac{1}{1296}x^{-16}y^{-12}z^{-20}}\right)^{2}$$

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

$$\frac{-2x^{2}z^{5}}{3y}\times \left(\frac{\frac{243}{4}x^{15}y^{22}z^{28}}{\frac{1}{1296}}\right)^{2}$$

Divide \frac{243}{4}x^{15}y^{22}z^{28} by \frac{1}{1296} by multiplying \frac{243}{4}x^{15}y^{22}z^{28} by the reciprocal of \frac{1}{1296}.

$$\frac{-2x^{2}z^{5}}{3y}\times \left(\frac{243}{4}x^{15}y^{22}z^{28}\times 1296\right)^{2}$$

Expand \left(\frac{243}{4}x^{15}y^{22}z^{28}\times 1296\right)^{2}.

$$\frac{-2x^{2}z^{5}}{3y}\times \left(\frac{243}{4}\right)^{2}\left(x^{15}\right)^{2}\left(y^{22}\right)^{2}\left(z^{28}\right)^{2}\times 1296^{2}$$

To raise a power to another power, multiply the exponents. Multiply 15 and 2 to get 30.

$$\frac{-2x^{2}z^{5}}{3y}\times \left(\frac{243}{4}\right)^{2}x^{30}\left(y^{22}\right)^{2}\left(z^{28}\right)^{2}\times 1296^{2}$$

To raise a power to another power, multiply the exponents. Multiply 22 and 2 to get 44.

$$\frac{-2x^{2}z^{5}}{3y}\times \left(\frac{243}{4}\right)^{2}x^{30}y^{44}\left(z^{28}\right)^{2}\times 1296^{2}$$

To raise a power to another power, multiply the exponents. Multiply 28 and 2 to get 56.

$$\frac{-2x^{2}z^{5}}{3y}\times \left(\frac{243}{4}\right)^{2}x^{30}y^{44}z^{56}\times 1296^{2}$$

Calculate \frac{243}{4} to the power of 2 and get \frac{59049}{16}.

$$\frac{-2x^{2}z^{5}}{3y}\times \frac{59049}{16}x^{30}y^{44}z^{56}\times 1296^{2}$$

Calculate 1296 to the power of 2 and get 1679616.

$$\frac{-2x^{2}z^{5}}{3y}\times \frac{59049}{16}x^{30}y^{44}z^{56}\times 1679616$$

Multiply \frac{59049}{16} and 1679616 to get 6198727824.

$$\frac{-2x^{2}z^{5}}{3y}\times 6198727824x^{30}y^{44}z^{56}$$

Express \frac{-2x^{2}z^{5}}{3y}\times 6198727824 as a single fraction.

$$\frac{-2x^{2}z^{5}\times 6198727824}{3y}x^{30}y^{44}z^{56}$$

Cancel out 3 in both numerator and denominator.

$$\frac{-2\times 2066242608x^{2}z^{5}}{y}x^{30}y^{44}z^{56}$$

Express \frac{-2\times 2066242608x^{2}z^{5}}{y}x^{30} as a single fraction.

$$\frac{-2\times 2066242608x^{2}z^{5}x^{30}}{y}y^{44}z^{56}$$

Express \frac{-2\times 2066242608x^{2}z^{5}x^{30}}{y}y^{44} as a single fraction.

$$\frac{-2\times 2066242608x^{2}z^{5}x^{30}y^{44}}{y}z^{56}$$

Cancel out y in both numerator and denominator.

$$-2\times 2066242608x^{2}z^{5}x^{30}y^{43}z^{56}$$

To multiply powers of the same base, add their exponents. Add 2 and 30 to get 32.

$$-2\times 2066242608x^{32}z^{5}y^{43}z^{56}$$

Multiply -2 and 2066242608 to get -4132485216.

$$-4132485216x^{32}z^{5}y^{43}z^{56}$$

To multiply powers of the same base, add their exponents. Add 5 and 56 to get 61.

$$-4132485216x^{32}z^{61}y^{43}$$