Solve 17x^2/51x^5

$$\frac { 17 x ^ { 2 } } { 51 x ^ { 5 } }$$

Evaluate

$$\frac{1}{3x^{3}}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(17x^{2}\right)^{1}\times \frac{1}{51x^{5}}$$

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

$$17^{1}\left(x^{2}\right)^{1}\times \frac{1}{51}\times \frac{1}{x^{5}}$$

Use the Commutative Property of Multiplication.

$$17^{1}\times \frac{1}{51}\left(x^{2}\right)^{1}\times \frac{1}{x^{5}}$$

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

$$17^{1}\times \frac{1}{51}x^{2}x^{5\left(-1\right)}$$

Multiply 5 times -1.

$$17^{1}\times \frac{1}{51}x^{2}x^{-5}$$

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

$$17^{1}\times \frac{1}{51}x^{2-5}$$

Add the exponents 2 and -5.

$$17^{1}\times \frac{1}{51}x^{-3}$$

Raise 17 to the power 1.

$$17\times \frac{1}{51}x^{-3}$$

Multiply 17 times \frac{1}{51}.

$$\frac{1}{3}x^{-3}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{17^{1}x^{2}}{51^{1}x^{5}}$$

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

$$\frac{17^{1}x^{2-5}}{51^{1}}$$

Subtract 5 from 2.

$$\frac{17^{1}x^{-3}}{51^{1}}$$

Reduce the fraction \frac{17}{51} to lowest terms by extracting and canceling out 17.

$$\frac{1}{3}x^{-3}$$

Differentiate w.r.t. x

$$-\frac{1}{x^{4}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{17}{51}x^{2-5})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{3}x^{-3})$$

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}.

$$-3\times \frac{1}{3}x^{-3-1}$$

Do the arithmetic.

$$-x^{-4}$$