Solve frac{8x^{18}y^7}{16x^8y^11})^2

$$\frac { 8 x ^ { 18 } y ^ { 7 } } { 16 x ^ { 8 } y ^ { 11 } } ) ^ { 2 }$$

Evaluate

$$\frac{x^{10}}{2y^{4}}$$

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{8^{1}x^{18}y^{7}}{16^{1}x^{8}y^{11}}$$

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

$$\frac{8^{1}}{16^{1}}x^{18-8}y^{7-11}$$

Subtract 8 from 18.

$$\frac{8^{1}}{16^{1}}x^{10}y^{7-11}$$

Subtract 11 from 7.

$$\frac{8^{1}}{16^{1}}x^{10}y^{-4}$$

Reduce the fraction \frac{8}{16} to lowest terms by extracting and canceling out 8.

$$\frac{1}{2}x^{10}\times \frac{1}{y^{4}}$$

Differentiate w.r.t. x

$$\frac{5x^{9}}{y^{4}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{8y^{7}}{16y^{11}}x^{18-8})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{2y^{4}}x^{10})$$

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}.

$$10\times \frac{1}{2y^{4}}x^{10-1}$$

Do the arithmetic.

$$\frac{5}{y^{4}}x^{9}$$