Solve {(-2x^3y)^4:[(1/2xy)(-4y)^2]}:(-x)^7=

$$\{ ( - 2 x ^ { 3 } y ) ^ { 4 } : [ ( \frac { 1 } { 2 } x y ) ( - 4 y ) ^ { 2 } ] \} : ( - x ) ^ { 7 } =$$

Evaluate

$$-2yx^{4}$$

Solution Steps

Expand \left(-2x^{3}y\right)^{4}.

$$\frac{\frac{\left(-2\right)^{4}\left(x^{3}\right)^{4}y^{4}}{\frac{1}{2}xy\left(-4y\right)^{2}}}{\left(-x\right)^{7}}$$

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

$$\frac{\frac{\left(-2\right)^{4}x^{12}y^{4}}{\frac{1}{2}xy\left(-4y\right)^{2}}}{\left(-x\right)^{7}}$$

Calculate -2 to the power of 4 and get 16.

$$\frac{\frac{16x^{12}y^{4}}{\frac{1}{2}xy\left(-4y\right)^{2}}}{\left(-x\right)^{7}}$$

Expand \left(-4y\right)^{2}.

$$\frac{\frac{16x^{12}y^{4}}{\frac{1}{2}xy\left(-4\right)^{2}y^{2}}}{\left(-x\right)^{7}}$$

Calculate -4 to the power of 2 and get 16.

$$\frac{\frac{16x^{12}y^{4}}{\frac{1}{2}xy\times 16y^{2}}}{\left(-x\right)^{7}}$$

Multiply \frac{1}{2} and 16 to get 8.

$$\frac{\frac{16x^{12}y^{4}}{8xyy^{2}}}{\left(-x\right)^{7}}$$

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

$$\frac{\frac{16x^{12}y^{4}}{8xy^{3}}}{\left(-x\right)^{7}}$$

Cancel out 8xy^{3} in both numerator and denominator.

$$\frac{2yx^{11}}{\left(-x\right)^{7}}$$

Expand \left(-x\right)^{7}.

$$\frac{2yx^{11}}{\left(-1\right)^{7}x^{7}}$$

Calculate -1 to the power of 7 and get -1.

$$\frac{2yx^{11}}{-x^{7}}$$

Cancel out x^{7} in both numerator and denominator.

$$\frac{2yx^{4}}{-1}$$

Anything divided by -1 gives its opposite.

$$-2yx^{4}$$

Expand

$$-2yx^{4}$$

Solution Steps

Expand \left(-2x^{3}y\right)^{4}.

$$\frac{\frac{\left(-2\right)^{4}\left(x^{3}\right)^{4}y^{4}}{\frac{1}{2}xy\left(-4y\right)^{2}}}{\left(-x\right)^{7}}$$

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

$$\frac{\frac{\left(-2\right)^{4}x^{12}y^{4}}{\frac{1}{2}xy\left(-4y\right)^{2}}}{\left(-x\right)^{7}}$$

Calculate -2 to the power of 4 and get 16.

$$\frac{\frac{16x^{12}y^{4}}{\frac{1}{2}xy\left(-4y\right)^{2}}}{\left(-x\right)^{7}}$$

Expand \left(-4y\right)^{2}.

$$\frac{\frac{16x^{12}y^{4}}{\frac{1}{2}xy\left(-4\right)^{2}y^{2}}}{\left(-x\right)^{7}}$$

Calculate -4 to the power of 2 and get 16.

$$\frac{\frac{16x^{12}y^{4}}{\frac{1}{2}xy\times 16y^{2}}}{\left(-x\right)^{7}}$$

Multiply \frac{1}{2} and 16 to get 8.

$$\frac{\frac{16x^{12}y^{4}}{8xyy^{2}}}{\left(-x\right)^{7}}$$

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

$$\frac{\frac{16x^{12}y^{4}}{8xy^{3}}}{\left(-x\right)^{7}}$$

Cancel out 8xy^{3} in both numerator and denominator.

$$\frac{2yx^{11}}{\left(-x\right)^{7}}$$

Expand \left(-x\right)^{7}.

$$\frac{2yx^{11}}{\left(-1\right)^{7}x^{7}}$$

Calculate -1 to the power of 7 and get -1.

$$\frac{2yx^{11}}{-x^{7}}$$

Cancel out x^{7} in both numerator and denominator.

$$\frac{2yx^{4}}{-1}$$

Anything divided by -1 gives its opposite.

$$-2yx^{4}$$