Solve 32d^7/8d^3

$$\frac { 32 d ^ { 7 } } { 8 d ^ { 3 } }$$

Evaluate

$$4d^{4}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(32d^{7}\right)^{1}\times \frac{1}{8d^{3}}$$

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

$$32^{1}\left(d^{7}\right)^{1}\times \frac{1}{8}\times \frac{1}{d^{3}}$$

Use the Commutative Property of Multiplication.

$$32^{1}\times \frac{1}{8}\left(d^{7}\right)^{1}\times \frac{1}{d^{3}}$$

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

$$32^{1}\times \frac{1}{8}d^{7}d^{3\left(-1\right)}$$

Multiply 3 times -1.

$$32^{1}\times \frac{1}{8}d^{7}d^{-3}$$

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

$$32^{1}\times \frac{1}{8}d^{7-3}$$

Add the exponents 7 and -3.

$$32^{1}\times \frac{1}{8}d^{4}$$

Raise 32 to the power 1.

$$32\times \frac{1}{8}d^{4}$$

Multiply 32 times \frac{1}{8}.

$$4d^{4}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{32^{1}d^{7}}{8^{1}d^{3}}$$

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

$$\frac{32^{1}d^{7-3}}{8^{1}}$$

Subtract 3 from 7.

$$\frac{32^{1}d^{4}}{8^{1}}$$

Divide 32 by 8.

$$4d^{4}$$

Differentiate w.r.t. d

$$16d^{3}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}d}(\frac{32}{8}d^{7-3})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}d}(4d^{4})$$

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}.

$$4\times 4d^{4-1}$$

Do the arithmetic.

$$16d^{3}$$