Solve x^{3}y^{-2}(x^-2y^-3/x-4y^3/x^2)

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

Evaluate

$$-4xy+\frac{1}{y^{5}}$$

Short Solution Steps

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x^{3} and x^{2} is x^{3}. Multiply \frac{4y^{3}}{x^{2}} times \frac{x}{x}.

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

Since \frac{y^{-3}}{x^{3}} and \frac{4y^{3}x}{x^{3}} have the same denominator, subtract them by subtracting their numerators.

$$x^{3}y^{-2}\times \frac{y^{-3}-4y^{3}x}{x^{3}}$$

Cancel out x^{3} and x^{3}.

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

Use the distributive property to multiply y^{-3}-4y^{3}x by y^{-2}.

$$y^{-5}-4y^{3}xy^{-2}$$

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

$$y^{-5}-4y^{1}x$$

Calculate y to the power of 1 and get y.

$$y^{-5}-4yx$$

Expand

$$-4xy+\frac{1}{y^{5}}$$

Short Solution Steps

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x^{3} and x^{2} is x^{3}. Multiply \frac{4y^{3}}{x^{2}} times \frac{x}{x}.

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

Since \frac{y^{-3}}{x^{3}} and \frac{4y^{3}x}{x^{3}} have the same denominator, subtract them by subtracting their numerators.

$$x^{3}y^{-2}\times \frac{y^{-3}-4y^{3}x}{x^{3}}$$

Cancel out x^{3} and x^{3}.

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

Use the distributive property to multiply y^{-3}-4y^{3}x by y^{-2}.

$$y^{-5}-4y^{3}xy^{-2}$$

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

$$y^{-5}-4y^{1}x$$

Calculate y to the power of 1 and get y.

$$y^{-5}-4yx$$