Solve 6x^3/3x^-5

$$\frac{ 6 { x }^{ 3 } }{ 3 { x }^{ -5 } }$$

Evaluate

$$2x^{8}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(6x^{3}\right)^{1}\times \frac{1}{3x^{-5}}$$

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

$$6^{1}\left(x^{3}\right)^{1}\times \frac{1}{3}\times \frac{1}{x^{-5}}$$

Use the Commutative Property of Multiplication.

$$6^{1}\times \frac{1}{3}\left(x^{3}\right)^{1}\times \frac{1}{x^{-5}}$$

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

$$6^{1}\times \frac{1}{3}x^{3}x^{-5\left(-1\right)}$$

Multiply -5 times -1.

$$6^{1}\times \frac{1}{3}x^{3}x^{5}$$

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

$$6^{1}\times \frac{1}{3}x^{3+5}$$

Add the exponents 3 and 5.

$$6^{1}\times \frac{1}{3}x^{8}$$

Raise 6 to the power 1.

$$6\times \frac{1}{3}x^{8}$$

Multiply 6 times \frac{1}{3}.

$$2x^{8}$$

Differentiate w.r.t. x

$$16x^{7}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{6}{3}x^{3-\left(-5\right)})$$

Do the arithmetic.

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

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}.

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

Do the arithmetic.

$$16x^{7}$$