Solve (3125t^125)^frac{1{3}}

$$( 3125 t ^ { 125 } ) ^ { \frac { 1 } { 3 } }$$

Evaluate

$$5^{\frac{5}{3}}t^{\frac{125}{3}}$$

Steps for Raising Power to Another Power

Use the rules of exponents to simplify the expression.

$$\sqrt[3]{3125t^{125}}$$

To raise the product of two or more numbers to a power, raise each number to the power and take their product.

$$\sqrt[3]{3125}\sqrt[3]{t^{125}}$$

Raise 3125 to the power \frac{1}{3}.

$$5\times 5^{\frac{2}{3}}\sqrt[3]{t^{125}}$$

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

$$5\times 5^{\frac{2}{3}}t^{125\times \frac{1}{3}}$$

Multiply 125 times \frac{1}{3}.

$$5\times 5^{\frac{2}{3}}t^{\frac{125}{3}}$$

Differentiate w.r.t. t

$$\frac{625\times 5^{\frac{2}{3}}t^{\frac{122}{3}}}{3}$$

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).

$$\frac{1}{3}\times \left(3125t^{125}\right)^{\frac{1}{3}-1}\frac{\mathrm{d}}{\mathrm{d}t}(3125t^{125})$$

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}.

$$\frac{1}{3}\times \left(3125t^{125}\right)^{-\frac{2}{3}}\times 125\times 3125t^{125-1}$$

Simplify.

$$\frac{390625}{3}t^{124}\times \left(3125t^{125}\right)^{-\frac{2}{3}}$$