Solve 4x^3y^15/4x^1y^5

$$\frac { 4 x ^ { 3 } y ^ { 15 } } { 4 x ^ { 1 } y ^ { 5 } }$$

Evaluate

$$x^{2}y^{10}$$

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{4^{1}x^{3}y^{15}}{4^{1}x^{1}y^{5}}$$

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

$$4^{1-1}x^{3-1}y^{15-5}$$

Subtract 1 from 1.

$$4^{0}x^{3-1}y^{15-5}$$

For any number a except 0, a^{0}=1.

$$x^{3-1}y^{15-5}$$

Subtract 1 from 3.

$$x^{2}y^{15-5}$$

Subtract 5 from 15.

$$x^{2}y^{10}$$

Differentiate w.r.t. x

$$2xy^{10}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{4y^{15}}{4y^{5}}x^{3-1})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}x}(y^{10}x^{2})$$

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}.

$$2y^{10}x^{2-1}$$

Do the arithmetic.

$$2y^{10}x^{1}$$

For any term t, t^{1}=t.

$$2y^{10}x$$