Solve (-1/2x^{2}yz)^{2}*(-4/3xy^2z)^2:(-1/9x^3y^2z)

$$( - \frac { 1 } { 2 } x ^ { 2 } y z ) ^ { 2 } \cdot ( - \frac { 4 } { 3 } x y ^ { 2 } z ) ^ { 2 } : ( - \frac { 1 } { 9 } x ^ { 3 } y ^ { 2 } z )$$

Evaluate

$$-4y^{4}\left(xz\right)^{3}$$

Solution Steps

Expand \left(-\frac{1}{2}x^{2}yz\right)^{2}.

$$\frac{\left(-\frac{1}{2}\right)^{2}\left(x^{2}\right)^{2}y^{2}z^{2}\left(-\frac{4}{3}xy^{2}z\right)^{2}}{-\frac{1}{9}x^{3}y^{2}z}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\frac{\left(-\frac{1}{2}\right)^{2}x^{4}y^{2}z^{2}\left(-\frac{4}{3}xy^{2}z\right)^{2}}{-\frac{1}{9}x^{3}y^{2}z}$$

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

$$\frac{\frac{1}{4}x^{4}y^{2}z^{2}\left(-\frac{4}{3}xy^{2}z\right)^{2}}{-\frac{1}{9}x^{3}y^{2}z}$$

Expand \left(-\frac{4}{3}xy^{2}z\right)^{2}.

$$\frac{\frac{1}{4}x^{4}y^{2}z^{2}\left(-\frac{4}{3}\right)^{2}x^{2}\left(y^{2}\right)^{2}z^{2}}{-\frac{1}{9}x^{3}y^{2}z}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\frac{\frac{1}{4}x^{4}y^{2}z^{2}\left(-\frac{4}{3}\right)^{2}x^{2}y^{4}z^{2}}{-\frac{1}{9}x^{3}y^{2}z}$$

Calculate -\frac{4}{3} to the power of 2 and get \frac{16}{9}.

$$\frac{\frac{1}{4}x^{4}y^{2}z^{2}\times \frac{16}{9}x^{2}y^{4}z^{2}}{-\frac{1}{9}x^{3}y^{2}z}$$

Multiply \frac{1}{4} and \frac{16}{9} to get \frac{4}{9}.

$$\frac{\frac{4}{9}x^{4}y^{2}z^{2}x^{2}y^{4}z^{2}}{-\frac{1}{9}x^{3}y^{2}z}$$

To multiply powers of the same base, add their exponents. Add 4 and 2 to get 6.

$$\frac{\frac{4}{9}x^{6}y^{2}z^{2}y^{4}z^{2}}{-\frac{1}{9}x^{3}y^{2}z}$$

To multiply powers of the same base, add their exponents. Add 2 and 4 to get 6.

$$\frac{\frac{4}{9}x^{6}y^{6}z^{2}z^{2}}{-\frac{1}{9}x^{3}y^{2}z}$$

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$\frac{\frac{4}{9}x^{6}y^{6}z^{4}}{-\frac{1}{9}x^{3}y^{2}z}$$

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

$$\frac{\frac{4}{9}x^{3}z^{3}y^{4}}{-\frac{1}{9}}$$

Divide \frac{4}{9}x^{3}z^{3}y^{4} by -\frac{1}{9} by multiplying \frac{4}{9}x^{3}z^{3}y^{4} by the reciprocal of -\frac{1}{9}.

$$\frac{\frac{4}{9}x^{3}z^{3}y^{4}\times 9}{-1}$$

Multiply \frac{4}{9} and 9 to get 4.

$$\frac{4x^{3}z^{3}y^{4}}{-1}$$

Anything divided by -1 gives its opposite.

$$-4x^{3}z^{3}y^{4}$$

Expand

$$-4y^{4}\left(xz\right)^{3}$$

Solution Steps

Expand \left(-\frac{1}{2}x^{2}yz\right)^{2}.

$$\frac{\left(-\frac{1}{2}\right)^{2}\left(x^{2}\right)^{2}y^{2}z^{2}\left(-\frac{4}{3}xy^{2}z\right)^{2}}{-\frac{1}{9}x^{3}y^{2}z}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\frac{\left(-\frac{1}{2}\right)^{2}x^{4}y^{2}z^{2}\left(-\frac{4}{3}xy^{2}z\right)^{2}}{-\frac{1}{9}x^{3}y^{2}z}$$

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

$$\frac{\frac{1}{4}x^{4}y^{2}z^{2}\left(-\frac{4}{3}xy^{2}z\right)^{2}}{-\frac{1}{9}x^{3}y^{2}z}$$

Expand \left(-\frac{4}{3}xy^{2}z\right)^{2}.

$$\frac{\frac{1}{4}x^{4}y^{2}z^{2}\left(-\frac{4}{3}\right)^{2}x^{2}\left(y^{2}\right)^{2}z^{2}}{-\frac{1}{9}x^{3}y^{2}z}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\frac{\frac{1}{4}x^{4}y^{2}z^{2}\left(-\frac{4}{3}\right)^{2}x^{2}y^{4}z^{2}}{-\frac{1}{9}x^{3}y^{2}z}$$

Calculate -\frac{4}{3} to the power of 2 and get \frac{16}{9}.

$$\frac{\frac{1}{4}x^{4}y^{2}z^{2}\times \frac{16}{9}x^{2}y^{4}z^{2}}{-\frac{1}{9}x^{3}y^{2}z}$$

Multiply \frac{1}{4} and \frac{16}{9} to get \frac{4}{9}.

$$\frac{\frac{4}{9}x^{4}y^{2}z^{2}x^{2}y^{4}z^{2}}{-\frac{1}{9}x^{3}y^{2}z}$$

To multiply powers of the same base, add their exponents. Add 4 and 2 to get 6.

$$\frac{\frac{4}{9}x^{6}y^{2}z^{2}y^{4}z^{2}}{-\frac{1}{9}x^{3}y^{2}z}$$

To multiply powers of the same base, add their exponents. Add 2 and 4 to get 6.

$$\frac{\frac{4}{9}x^{6}y^{6}z^{2}z^{2}}{-\frac{1}{9}x^{3}y^{2}z}$$

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$\frac{\frac{4}{9}x^{6}y^{6}z^{4}}{-\frac{1}{9}x^{3}y^{2}z}$$

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

$$\frac{\frac{4}{9}x^{3}z^{3}y^{4}}{-\frac{1}{9}}$$

Divide \frac{4}{9}x^{3}z^{3}y^{4} by -\frac{1}{9} by multiplying \frac{4}{9}x^{3}z^{3}y^{4} by the reciprocal of -\frac{1}{9}.

$$\frac{\frac{4}{9}x^{3}z^{3}y^{4}\times 9}{-1}$$

Multiply \frac{4}{9} and 9 to get 4.

$$\frac{4x^{3}z^{3}y^{4}}{-1}$$

Anything divided by -1 gives its opposite.

$$-4x^{3}z^{3}y^{4}$$