Solve 99x^13/198x^19

$$\frac{ 99 { x }^{ 13 } }{ 198 { x }^{ 19 } }$$

Evaluate

$$\frac{1}{2x^{6}}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(99x^{13}\right)^{1}\times \frac{1}{198x^{19}}$$

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

$$99^{1}\left(x^{13}\right)^{1}\times \frac{1}{198}\times \frac{1}{x^{19}}$$

Use the Commutative Property of Multiplication.

$$99^{1}\times \frac{1}{198}\left(x^{13}\right)^{1}\times \frac{1}{x^{19}}$$

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

$$99^{1}\times \frac{1}{198}x^{13}x^{19\left(-1\right)}$$

Multiply 19 times -1.

$$99^{1}\times \frac{1}{198}x^{13}x^{-19}$$

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

$$99^{1}\times \frac{1}{198}x^{13-19}$$

Add the exponents 13 and -19.

$$99^{1}\times \frac{1}{198}x^{-6}$$

Raise 99 to the power 1.

$$99\times \frac{1}{198}x^{-6}$$

Multiply 99 times \frac{1}{198}.

$$\frac{1}{2}x^{-6}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{99^{1}x^{13}}{198^{1}x^{19}}$$

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

$$\frac{99^{1}x^{13-19}}{198^{1}}$$

Subtract 19 from 13.

$$\frac{99^{1}x^{-6}}{198^{1}}$$

Reduce the fraction \frac{99}{198} to lowest terms by extracting and canceling out 99.

$$\frac{1}{2}x^{-6}$$

Differentiate w.r.t. x

$$-\frac{3}{x^{7}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{99}{198}x^{13-19})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{2}x^{-6})$$

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}.

$$-6\times \frac{1}{2}x^{-6-1}$$

Do the arithmetic.

$$-3x^{-7}$$