Solve (frac{8x^{3}y^{-2}z^{4}}{4x^-2y^4z^-3})^-2

$$( \frac { 8 x ^ { 3 } y ^ { - 2 } z ^ { 4 } } { 4 x ^ { - 2 } y ^ { 4 } z ^ { - 3 } } ) ^ { - 2 }$$

Evaluate

$$\frac{y^{12}}{4x^{10}z^{14}}$$

Solution Steps

Cancel out 4 in both numerator and denominator.

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

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

$$\left(\frac{2y^{-2}x^{5}z^{7}}{y^{4}}\right)^{-2}$$

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

$$\left(\frac{2x^{5}z^{7}}{y^{6}}\right)^{-2}$$

To raise \frac{2x^{5}z^{7}}{y^{6}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(2x^{5}z^{7}\right)^{-2}}{\left(y^{6}\right)^{-2}}$$

To raise a power to another power, multiply the exponents. Multiply 6 and -2 to get -12.

$$\frac{\left(2x^{5}z^{7}\right)^{-2}}{y^{-12}}$$

Expand \left(2x^{5}z^{7}\right)^{-2}.

$$\frac{2^{-2}\left(x^{5}\right)^{-2}\left(z^{7}\right)^{-2}}{y^{-12}}$$

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

$$\frac{2^{-2}x^{-10}\left(z^{7}\right)^{-2}}{y^{-12}}$$

To raise a power to another power, multiply the exponents. Multiply 7 and -2 to get -14.

$$\frac{2^{-2}x^{-10}z^{-14}}{y^{-12}}$$

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

$$\frac{\frac{1}{4}x^{-10}z^{-14}}{y^{-12}}$$

Expand

$$\frac{y^{12}}{4x^{10}z^{14}}$$

Solution Steps

Cancel out 4 in both numerator and denominator.

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

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

$$\left(\frac{2y^{-2}x^{5}z^{7}}{y^{4}}\right)^{-2}$$

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

$$\left(\frac{2x^{5}z^{7}}{y^{6}}\right)^{-2}$$

To raise \frac{2x^{5}z^{7}}{y^{6}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(2x^{5}z^{7}\right)^{-2}}{\left(y^{6}\right)^{-2}}$$

To raise a power to another power, multiply the exponents. Multiply 6 and -2 to get -12.

$$\frac{\left(2x^{5}z^{7}\right)^{-2}}{y^{-12}}$$

Expand \left(2x^{5}z^{7}\right)^{-2}.

$$\frac{2^{-2}\left(x^{5}\right)^{-2}\left(z^{7}\right)^{-2}}{y^{-12}}$$

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

$$\frac{2^{-2}x^{-10}\left(z^{7}\right)^{-2}}{y^{-12}}$$

To raise a power to another power, multiply the exponents. Multiply 7 and -2 to get -14.

$$\frac{2^{-2}x^{-10}z^{-14}}{y^{-12}}$$

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

$$\frac{\frac{1}{4}x^{-10}z^{-14}}{y^{-12}}$$