Solve (x^5)^5(x^3)^4

$$( x ^ { 5 } ) ^ { 5 } ( x ^ { 3 } ) ^ { 4 }$$

Evaluate

$$x^{37}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(x^{5}\right)^{5}\left(x^{3}\right)^{4}$$

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

$$x^{5\times 5}x^{3\times 4}$$

Multiply 5 times 5.

$$x^{25}x^{3\times 4}$$

Multiply 3 times 4.

$$x^{25}x^{12}$$

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

$$x^{25+12}$$

Add the exponents 25 and 12.

$$x^{37}$$

Differentiate w.r.t. x

$$37x^{36}$$

Steps Using Definition of a Derivative

To raise a power to another power, multiply the exponents. Multiply 5 and 5 to get 25.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{25}\left(x^{3}\right)^{4})$$

To raise a power to another power, multiply the exponents. Multiply 3 and 4 to get 12.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{25}x^{12})$$

To multiply powers of the same base, add their exponents. Add 25 and 12 to get 37.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{37})$$

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

$$37x^{37-1}$$

Subtract 1 from 37.

$$37x^{36}$$