Solve 12t^6/4t^3

$$\frac { 12 t ^ { 6 } } { 4 t ^ { 3 } }$$

Evaluate

$$3t^{3}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(12t^{6}\right)^{1}\times \frac{1}{4t^{3}}$$

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

$$12^{1}\left(t^{6}\right)^{1}\times \frac{1}{4}\times \frac{1}{t^{3}}$$

Use the Commutative Property of Multiplication.

$$12^{1}\times \frac{1}{4}\left(t^{6}\right)^{1}\times \frac{1}{t^{3}}$$

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

$$12^{1}\times \frac{1}{4}t^{6}t^{3\left(-1\right)}$$

Multiply 3 times -1.

$$12^{1}\times \frac{1}{4}t^{6}t^{-3}$$

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

$$12^{1}\times \frac{1}{4}t^{6-3}$$

Add the exponents 6 and -3.

$$12^{1}\times \frac{1}{4}t^{3}$$

Raise 12 to the power 1.

$$12\times \frac{1}{4}t^{3}$$

Multiply 12 times \frac{1}{4}.

$$3t^{3}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{12^{1}t^{6}}{4^{1}t^{3}}$$

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

$$\frac{12^{1}t^{6-3}}{4^{1}}$$

Subtract 3 from 6.

$$\frac{12^{1}t^{3}}{4^{1}}$$

Divide 12 by 4.

$$3t^{3}$$

Differentiate w.r.t. t

$$9t^{2}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}t}(\frac{12}{4}t^{6-3})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}t}(3t^{3})$$

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}.

$$3\times 3t^{3-1}$$

Do the arithmetic.

$$9t^{2}$$