Solve 2x^2/8x^8

$$\frac { 2 x ^ { 2 } } { 8 x ^ { 8 } }$$

Evaluate

$$\frac{1}{4x^{6}}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(2x^{2}\right)^{1}\times \frac{1}{8x^{8}}$$

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

$$2^{1}\left(x^{2}\right)^{1}\times \frac{1}{8}\times \frac{1}{x^{8}}$$

Use the Commutative Property of Multiplication.

$$2^{1}\times \frac{1}{8}\left(x^{2}\right)^{1}\times \frac{1}{x^{8}}$$

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

$$2^{1}\times \frac{1}{8}x^{2}x^{8\left(-1\right)}$$

Multiply 8 times -1.

$$2^{1}\times \frac{1}{8}x^{2}x^{-8}$$

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

$$2^{1}\times \frac{1}{8}x^{2-8}$$

Add the exponents 2 and -8.

$$2^{1}\times \frac{1}{8}x^{-6}$$

Raise 2 to the power 1.

$$2\times \frac{1}{8}x^{-6}$$

Multiply 2 times \frac{1}{8}.

$$\frac{1}{4}x^{-6}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{2^{1}x^{2}}{8^{1}x^{8}}$$

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

$$\frac{2^{1}x^{2-8}}{8^{1}}$$

Subtract 8 from 2.

$$\frac{2^{1}x^{-6}}{8^{1}}$$

Reduce the fraction \frac{2}{8} to lowest terms by extracting and canceling out 2.

$$\frac{1}{4}x^{-6}$$

Differentiate w.r.t. x

$$-\frac{3}{2x^{7}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2}{8}x^{2-8})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{4}x^{-6})$$

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}.

$$-6\times \frac{1}{4}x^{-6-1}$$

Do the arithmetic.

$$-\frac{3}{2}x^{-7}$$