Solve 54y^10/6y^5

$$\frac { 54 y ^ { 10 } } { 6 y ^ { 5 } }$$

Evaluate

$$9y^{5}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\left(54y^{10}\right)^{1}\times \frac{1}{6y^{5}}$$

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

$$54^{1}\left(y^{10}\right)^{1}\times \frac{1}{6}\times \frac{1}{y^{5}}$$

Use the Commutative Property of Multiplication.

$$54^{1}\times \frac{1}{6}\left(y^{10}\right)^{1}\times \frac{1}{y^{5}}$$

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

$$54^{1}\times \frac{1}{6}y^{10}y^{5\left(-1\right)}$$

Multiply 5 times -1.

$$54^{1}\times \frac{1}{6}y^{10}y^{-5}$$

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

$$54^{1}\times \frac{1}{6}y^{10-5}$$

Add the exponents 10 and -5.

$$54^{1}\times \frac{1}{6}y^{5}$$

Raise 54 to the power 1.

$$54\times \frac{1}{6}y^{5}$$

Multiply 54 times \frac{1}{6}.

$$9y^{5}$$

Steps for Quotient of Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{54^{1}y^{10}}{6^{1}y^{5}}$$

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

$$\frac{54^{1}y^{10-5}}{6^{1}}$$

Subtract 5 from 10.

$$\frac{54^{1}y^{5}}{6^{1}}$$

Divide 54 by 6.

$$9y^{5}$$

Differentiate w.r.t. y

$$45y^{4}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}y}(\frac{54}{6}y^{10-5})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}y}(9y^{5})$$

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}.

$$5\times 9y^{5-1}$$

Do the arithmetic.

$$45y^{4}$$