Solve yfrac{(-2)^{10}*5^{6}*6^3}{(-2)^6*5^3*6^2}

$$y \frac { ( - 2 ) ^ { 10 } \cdot 5 ^ { 6 } \cdot 6 ^ { 3 } } { ( - 2 ) ^ { 6 } \cdot 5 ^ { 3 } \cdot 6 ^ { 2 } }$$

Evaluate

$$12000y$$

Solution Steps

Cancel out \left(-2\right)^{6}\times 5^{3}\times 6^{2} in both numerator and denominator.

$$y\times 6\left(-2\right)^{4}\times 5^{3}$$

Calculate -2 to the power of 4 and get 16.

$$y\times 6\times 16\times 5^{3}$$

Multiply 6 and 16 to get 96.

$$y\times 96\times 5^{3}$$

Calculate 5 to the power of 3 and get 125.

$$y\times 96\times 125$$

Multiply 96 and 125 to get 12000.

$$y\times 12000$$

Differentiate w.r.t. y

$$12000$$

Steps Using Definition of a Derivative

Cancel out \left(-2\right)^{6}\times 5^{3}\times 6^{2} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}y}(y\times 6\left(-2\right)^{4}\times 5^{3})$$

Calculate -2 to the power of 4 and get 16.

$$\frac{\mathrm{d}}{\mathrm{d}y}(y\times 6\times 16\times 5^{3})$$

Multiply 6 and 16 to get 96.

$$\frac{\mathrm{d}}{\mathrm{d}y}(y\times 96\times 5^{3})$$

Calculate 5 to the power of 3 and get 125.

$$\frac{\mathrm{d}}{\mathrm{d}y}(y\times 96\times 125)$$

Multiply 96 and 125 to get 12000.

$$\frac{\mathrm{d}}{\mathrm{d}y}(y\times 12000)$$

The derivative of ax^{n} is nax^{n-1}.

$$12000y^{1-1}$$

Subtract 1 from 1.

$$12000y^{0}$$

For any term t except 0, t^{0}=1.

$$12000\times 1$$

For any term t, t\times 1=t and 1t=t.

$$12000$$