Solve 27x^5/36x^7

$$\frac { 27 x ^ { 5 } } { 36 x ^ { 7 } }$$

Evaluate

$$\frac{3}{4x^{2}}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(27x^{5}\right)^{1}\times \frac{1}{36x^{7}}$$

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

$$27^{1}\left(x^{5}\right)^{1}\times \frac{1}{36}\times \frac{1}{x^{7}}$$

Use the Commutative Property of Multiplication.

$$27^{1}\times \frac{1}{36}\left(x^{5}\right)^{1}\times \frac{1}{x^{7}}$$

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

$$27^{1}\times \frac{1}{36}x^{5}x^{7\left(-1\right)}$$

Multiply 7 times -1.

$$27^{1}\times \frac{1}{36}x^{5}x^{-7}$$

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

$$27^{1}\times \frac{1}{36}x^{5-7}$$

Add the exponents 5 and -7.

$$27^{1}\times \frac{1}{36}x^{-2}$$

Raise 27 to the power 1.

$$27\times \frac{1}{36}x^{-2}$$

Multiply 27 times \frac{1}{36}.

$$\frac{3}{4}x^{-2}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{27^{1}x^{5}}{36^{1}x^{7}}$$

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

$$\frac{27^{1}x^{5-7}}{36^{1}}$$

Subtract 7 from 5.

$$\frac{27^{1}x^{-2}}{36^{1}}$$

Reduce the fraction \frac{27}{36} to lowest terms by extracting and canceling out 9.

$$\frac{3}{4}x^{-2}$$

Differentiate w.r.t. x

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

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{27}{36}x^{5-7})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{3}{4}x^{-2})$$

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}.

$$-2\times \frac{3}{4}x^{-2-1}$$

Do the arithmetic.

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