Solve 25x^6/5x^18=

$$\frac { 25 x ^ { 6 } } { 5 x ^ { 18 } } =$$

Evaluate

$$\frac{5}{x^{12}}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(25x^{6}\right)^{1}\times \frac{1}{5x^{18}}$$

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

$$25^{1}\left(x^{6}\right)^{1}\times \frac{1}{5}\times \frac{1}{x^{18}}$$

Use the Commutative Property of Multiplication.

$$25^{1}\times \frac{1}{5}\left(x^{6}\right)^{1}\times \frac{1}{x^{18}}$$

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

$$25^{1}\times \frac{1}{5}x^{6}x^{18\left(-1\right)}$$

Multiply 18 times -1.

$$25^{1}\times \frac{1}{5}x^{6}x^{-18}$$

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

$$25^{1}\times \frac{1}{5}x^{6-18}$$

Add the exponents 6 and -18.

$$25^{1}\times \frac{1}{5}x^{-12}$$

Raise 25 to the power 1.

$$25\times \frac{1}{5}x^{-12}$$

Multiply 25 times \frac{1}{5}.

$$5x^{-12}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{25^{1}x^{6}}{5^{1}x^{18}}$$

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

$$\frac{25^{1}x^{6-18}}{5^{1}}$$

Subtract 18 from 6.

$$\frac{25^{1}x^{-12}}{5^{1}}$$

Divide 25 by 5.

$$5x^{-12}$$

Differentiate w.r.t. x

$$-\frac{60}{x^{13}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{25}{5}x^{6-18})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(5x^{-12})$$

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}.

$$-12\times 5x^{-12-1}$$

Do the arithmetic.

$$-60x^{-13}$$