Solve frac{(8x^{5}y^{7})^-2}{4^2x^-8y^5}

$$\frac { ( 8 x ^ { 5 } y ^ { 7 } ) ^ { - 2 } } { 4 ^ { 2 } x ^ { - 8 } y ^ { 5 } }$$

Evaluate

$$\frac{1}{1024x^{2}y^{19}}$$

Solution Steps

Expand \left(8x^{5}y^{7}\right)^{-2}.

$$\frac{8^{-2}\left(x^{5}\right)^{-2}\left(y^{7}\right)^{-2}}{4^{2}x^{-8}y^{5}}$$

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

$$\frac{8^{-2}x^{-10}\left(y^{7}\right)^{-2}}{4^{2}x^{-8}y^{5}}$$

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

$$\frac{8^{-2}x^{-10}y^{-14}}{4^{2}x^{-8}y^{5}}$$

Calculate 8 to the power of -2 and get \frac{1}{64}.

$$\frac{\frac{1}{64}x^{-10}y^{-14}}{4^{2}x^{-8}y^{5}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{\frac{1}{64}x^{-10}y^{-14}}{16x^{-8}y^{5}}$$

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

$$\frac{\frac{1}{64}}{16x^{2}y^{19}}$$

Express \frac{\frac{1}{64}}{16x^{2}y^{19}} as a single fraction.

$$\frac{1}{64\times 16x^{2}y^{19}}$$

Multiply 64 and 16 to get 1024.

$$\frac{1}{1024x^{2}y^{19}}$$

Expand

$$\frac{1}{1024x^{2}y^{19}}$$

Solution Steps

Expand \left(8x^{5}y^{7}\right)^{-2}.

$$\frac{8^{-2}\left(x^{5}\right)^{-2}\left(y^{7}\right)^{-2}}{4^{2}x^{-8}y^{5}}$$

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

$$\frac{8^{-2}x^{-10}\left(y^{7}\right)^{-2}}{4^{2}x^{-8}y^{5}}$$

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

$$\frac{8^{-2}x^{-10}y^{-14}}{4^{2}x^{-8}y^{5}}$$

Calculate 8 to the power of -2 and get \frac{1}{64}.

$$\frac{\frac{1}{64}x^{-10}y^{-14}}{4^{2}x^{-8}y^{5}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{\frac{1}{64}x^{-10}y^{-14}}{16x^{-8}y^{5}}$$

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

$$\frac{\frac{1}{64}}{16x^{2}y^{19}}$$

Express \frac{\frac{1}{64}}{16x^{2}y^{19}} as a single fraction.

$$\frac{1}{64\times 16x^{2}y^{19}}$$

Multiply 64 and 16 to get 1024.

$$\frac{1}{1024x^{2}y^{19}}$$