Solve 30t^5/6t^2

$$\frac { 30 t ^ { 5 } } { 6 t ^ { 2 } }$$

Evaluate

$$5t^{3}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(30t^{5}\right)^{1}\times \frac{1}{6t^{2}}$$

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

$$30^{1}\left(t^{5}\right)^{1}\times \frac{1}{6}\times \frac{1}{t^{2}}$$

Use the Commutative Property of Multiplication.

$$30^{1}\times \frac{1}{6}\left(t^{5}\right)^{1}\times \frac{1}{t^{2}}$$

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

$$30^{1}\times \frac{1}{6}t^{5}t^{2\left(-1\right)}$$

Multiply 2 times -1.

$$30^{1}\times \frac{1}{6}t^{5}t^{-2}$$

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

$$30^{1}\times \frac{1}{6}t^{5-2}$$

Add the exponents 5 and -2.

$$30^{1}\times \frac{1}{6}t^{3}$$

Raise 30 to the power 1.

$$30\times \frac{1}{6}t^{3}$$

Multiply 30 times \frac{1}{6}.

$$5t^{3}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{30^{1}t^{5}}{6^{1}t^{2}}$$

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

$$\frac{30^{1}t^{5-2}}{6^{1}}$$

Subtract 2 from 5.

$$\frac{30^{1}t^{3}}{6^{1}}$$

Divide 30 by 6.

$$5t^{3}$$

Differentiate w.r.t. t

$$15t^{2}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}t}(\frac{30}{6}t^{5-2})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}t}(5t^{3})$$

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}.

$$3\times 5t^{3-1}$$

Do the arithmetic.

$$15t^{2}$$