Solve 27x^-3/9x^5

$$\frac { 27 x ^ { - 3 } } { 9 x ^ { 5 } }$$

Evaluate

$$\frac{3}{x^{8}}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(27x^{-3}\right)^{1}\times \frac{1}{9x^{5}}$$

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

$$27^{1}\left(x^{-3}\right)^{1}\times \frac{1}{9}\times \frac{1}{x^{5}}$$

Use the Commutative Property of Multiplication.

$$27^{1}\times \frac{1}{9}\left(x^{-3}\right)^{1}\times \frac{1}{x^{5}}$$

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

$$27^{1}\times \frac{1}{9}x^{-3}x^{5\left(-1\right)}$$

Multiply 5 times -1.

$$27^{1}\times \frac{1}{9}x^{-3}x^{-5}$$

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

$$27^{1}\times \frac{1}{9}x^{-3-5}$$

Add the exponents -3 and -5.

$$27^{1}\times \frac{1}{9}x^{-8}$$

Raise 27 to the power 1.

$$27\times \frac{1}{9}x^{-8}$$

Multiply 27 times \frac{1}{9}.

$$3x^{-8}$$

Differentiate w.r.t. x

$$-\frac{24}{x^{9}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{27}{9}x^{-3-5})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(3x^{-8})$$

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}.

$$-8\times 3x^{-8-1}$$

Do the arithmetic.

$$-24x^{-9}$$