Solve frac{{y}^frac{3{7}}}{{y}^frac{5{7}}}divfrac{{y}^frac{5{3}}}{{y}^frac{9{7}}}*y^2

$$\frac{ { y }^{ \frac{ 3 }{ 7 } } }{ { y }^{ \frac{ 5 }{ 7 } } } \div \frac { { y }^{ \frac{ 5 }{ 3 } } } { { y }^{ \frac{ 9 }{ 7 } } } \times { y }^{ 2 }$$

Evaluate

$$y^{\frac{4}{3}}$$

Steps Using Quotient of Powers Property

Rewrite y^{\frac{5}{7}} as y^{\frac{3}{7}}y^{\frac{2}{7}}. Cancel out y^{\frac{3}{7}} in both numerator and denominator.

$$\frac{\frac{1}{y^{\frac{2}{7}}}}{\frac{y^{\frac{5}{3}}}{y^{\frac{9}{7}}}}y^{2}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract \frac{9}{7} from \frac{5}{3} to get \frac{8}{21}.

$$\frac{\frac{1}{y^{\frac{2}{7}}}}{y^{\frac{8}{21}}}y^{2}$$

Express \frac{\frac{1}{y^{\frac{2}{7}}}}{y^{\frac{8}{21}}} as a single fraction.

$$\frac{1}{y^{\frac{2}{7}}y^{\frac{8}{21}}}y^{2}$$

To multiply powers of the same base, add their exponents. Add \frac{2}{7} and \frac{8}{21} to get \frac{2}{3}.

$$\frac{1}{y^{\frac{2}{3}}}y^{2}$$

Use the rules of exponents to simplify the expression.

$$\frac{y^{2}}{y^{\frac{2}{3}}}$$

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

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

Subtract \frac{2}{3} from 2.

$$y^{\frac{4}{3}}$$

Differentiate w.r.t. y

$$\frac{4\sqrt[3]{y}}{3}$$

Solution Steps

Rewrite y^{\frac{5}{7}} as y^{\frac{3}{7}}y^{\frac{2}{7}}. Cancel out y^{\frac{3}{7}} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{\frac{1}{y^{\frac{2}{7}}}}{\frac{y^{\frac{5}{3}}}{y^{\frac{9}{7}}}}y^{2})$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract \frac{9}{7} from \frac{5}{3} to get \frac{8}{21}.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{\frac{1}{y^{\frac{2}{7}}}}{y^{\frac{8}{21}}}y^{2})$$

Express \frac{\frac{1}{y^{\frac{2}{7}}}}{y^{\frac{8}{21}}} as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{1}{y^{\frac{2}{7}}y^{\frac{8}{21}}}y^{2})$$

To multiply powers of the same base, add their exponents. Add \frac{2}{7} and \frac{8}{21} to get \frac{2}{3}.

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{1}{y^{\frac{2}{3}}}y^{2})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{1}{1}y^{2-\frac{2}{3}})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}y}(y^{\frac{4}{3}})$$

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}.

$$\frac{4}{3}y^{\frac{4}{3}-1}$$

Do the arithmetic.

$$\frac{4}{3}\sqrt[3]{y}$$