Solve (y^5)^2(y^4)^4

$$( y ^ { 5 } ) ^ { 2 } ( y ^ { 4 } ) ^ { 4 }$$

Evaluate

$$y^{26}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(y^{5}\right)^{2}\left(y^{4}\right)^{4}$$

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

$$y^{5\times 2}y^{4\times 4}$$

Multiply 5 times 2.

$$y^{10}y^{4\times 4}$$

Multiply 4 times 4.

$$y^{10}y^{16}$$

To multiply powers of the same base, add their exponents.

$$y^{10+16}$$

Add the exponents 10 and 16.

$$y^{26}$$

Differentiate w.r.t. y

$$26y^{25}$$

Steps Using Definition of a Derivative

To raise a power to another power, multiply the exponents. Multiply 5 and 2 to get 10.

$$\frac{\mathrm{d}}{\mathrm{d}y}(y^{10}\left(y^{4}\right)^{4})$$

To raise a power to another power, multiply the exponents. Multiply 4 and 4 to get 16.

$$\frac{\mathrm{d}}{\mathrm{d}y}(y^{10}y^{16})$$

To multiply powers of the same base, add their exponents. Add 10 and 16 to get 26.

$$\frac{\mathrm{d}}{\mathrm{d}y}(y^{26})$$

The derivative of ax^{n} is nax^{n-1}.

$$26y^{26-1}$$

Subtract 1 from 26.

$$26y^{25}$$