Solve (2yx^{3})^4(49x^6y^2)^frac{1{2}}

$$( 2 y x ^ { 3 } ) ^ { 4 } ( 49 x ^ { 6 } y ^ { 2 } ) ^ { \frac { 1 } { 2 } }$$

Evaluate

$$112|y|\left(|x|\right)^{3}y^{4}x^{12}$$

Solution Steps

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

$$2^{4}y^{4}\left(x^{3}\right)^{4}\times \left(49x^{6}y^{2}\right)^{\frac{1}{2}}$$

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

$$2^{4}y^{4}x^{12}\times \left(49x^{6}y^{2}\right)^{\frac{1}{2}}$$

Calculate 2 to the power of 4 and get 16.

$$16y^{4}x^{12}\times \left(49x^{6}y^{2}\right)^{\frac{1}{2}}$$

Expand \left(49x^{6}y^{2}\right)^{\frac{1}{2}}.

$$16y^{4}x^{12}\times 49^{\frac{1}{2}}\left(x^{6}\right)^{\frac{1}{2}}\left(y^{2}\right)^{\frac{1}{2}}$$

To raise a power to another power, multiply the exponents. Multiply 6 and \frac{1}{2} to get 3.

$$16y^{4}x^{12}\times 49^{\frac{1}{2}}x^{3}\left(y^{2}\right)^{\frac{1}{2}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and \frac{1}{2} to get 1.

$$16y^{4}x^{12}\times 49^{\frac{1}{2}}x^{3}y^{1}$$

Calculate 49 to the power of \frac{1}{2} and get 7.

$$16y^{4}x^{12}\times 7x^{3}y^{1}$$

Calculate y to the power of 1 and get y.

$$16y^{4}x^{12}\times 7x^{3}y$$

Multiply 16 and 7 to get 112.

$$112y^{4}x^{12}x^{3}y$$

To multiply powers of the same base, add their exponents. Add 12 and 3 to get 15.

$$112y^{4}x^{15}y$$

To multiply powers of the same base, add their exponents. Add 4 and 1 to get 5.

$$112y^{5}x^{15}$$

Differentiate w.r.t. x

$$1680|x||y|y^{4}x^{13}$$