Solve (t^frac{6{14}})^frac{5{15}}=

$$( t ^ { \frac { 6 } { 14 } } ) ^ { \frac { 5 } { 15 } } =$$

Evaluate

$$\sqrt[7]{t}$$

Steps for Raising Power to Another Power

Use the rules of exponents to simplify the expression.

$$\sqrt[3]{t^{\frac{3}{7}}}$$

To raise a power to another power, multiply the exponents.

$$t^{\frac{3}{7}\times \frac{1}{3}}$$

Multiply \frac{3}{7} times \frac{1}{3} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$\sqrt[7]{t}$$

Differentiate w.r.t. t

$$\frac{1}{7t^{\frac{6}{7}}}$$

Steps Using Chain Rule

If F is the composition of two differentiable functions f\left(u\right) and u=g\left(x\right), that is, if F\left(x\right)=f\left(g\left(x\right)\right), then the derivative of F is the derivative of f with respect to u times the derivative of g with respect to x, that is, \frac{\mathrm{d}}{\mathrm{d}x}(F)\left(x\right)=\frac{\mathrm{d}}{\mathrm{d}x}(f)\left(g\left(x\right)\right)\frac{\mathrm{d}}{\mathrm{d}x}(g)\left(x\right).

$$\frac{1}{3}\left(t^{\frac{3}{7}}\right)^{\frac{1}{3}-1}\frac{\mathrm{d}}{\mathrm{d}t}(t^{\frac{3}{7}})$$

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}.

$$\frac{1}{3}\left(t^{\frac{3}{7}}\right)^{-\frac{2}{3}}\times \frac{3}{7}t^{\frac{3}{7}-1}$$

Simplify.

$$\frac{1}{7}t^{-\frac{4}{7}}\left(t^{\frac{3}{7}}\right)^{-\frac{2}{3}}$$