Solve frac{3^{-3}x^{2}y^-5}{5^-2x^7y^-2}

$$\frac { 3 ^ { - 3 } x ^ { 2 } y ^ { - 5 } } { 5 ^ { - 2 } x ^ { 7 } y ^ { - 2 } }$$

Evaluate

$$\frac{25}{27y^{3}x^{5}}$$

Solution Steps

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

$$\frac{3^{-3}y^{-5}}{5^{-2}y^{-2}x^{5}}$$

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

$$\frac{3^{-3}}{5^{-2}y^{3}x^{5}}$$

Calculate 3 to the power of -3 and get \frac{1}{27}.

$$\frac{\frac{1}{27}}{5^{-2}y^{3}x^{5}}$$

Calculate 5 to the power of -2 and get \frac{1}{25}.

$$\frac{\frac{1}{27}}{\frac{1}{25}y^{3}x^{5}}$$

Express \frac{\frac{1}{27}}{\frac{1}{25}y^{3}x^{5}} as a single fraction.

$$\frac{1}{27\times \frac{1}{25}y^{3}x^{5}}$$

Multiply 27 and \frac{1}{25} to get \frac{27}{25}.

$$\frac{1}{\frac{27}{25}y^{3}x^{5}}$$

Differentiate w.r.t. x

$$-\frac{125}{27y^{3}x^{6}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{27y^{5}\times \frac{1}{25y^{2}}}x^{2-7})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{25}{27y^{3}}x^{-5})$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$-5\times \frac{25}{27y^{3}}x^{-5-1}$$

Do the arithmetic.

$$\left(-\frac{125}{27y^{3}}\right)x^{-6}$$