Solve frac{4^{3}*25^{3}}{(3^2)^3*(2^3)^4}quadd

$$\frac { 4 ^ { 3 } \cdot 25 ^ { 3 } } { ( 3 ^ { 2 } ) ^ { 3 } \cdot ( 2 ^ { 3 } ) ^ { 4 } } \quad d$$

Evaluate

$$\frac{15625d}{46656}$$

Solution Steps

To raise a power to another power, multiply the exponents. Multiply 2 and 3 to get 6.

$$\frac{4^{3}\times 25^{3}}{3^{6}\times \left(2^{3}\right)^{4}}d$$

To raise a power to another power, multiply the exponents. Multiply 3 and 4 to get 12.

$$\frac{4^{3}\times 25^{3}}{3^{6}\times 2^{12}}d$$

Calculate 4 to the power of 3 and get 64.

$$\frac{64\times 25^{3}}{3^{6}\times 2^{12}}d$$

Calculate 25 to the power of 3 and get 15625.

$$\frac{64\times 15625}{3^{6}\times 2^{12}}d$$

Multiply 64 and 15625 to get 1000000.

$$\frac{1000000}{3^{6}\times 2^{12}}d$$

Calculate 3 to the power of 6 and get 729.

$$\frac{1000000}{729\times 2^{12}}d$$

Calculate 2 to the power of 12 and get 4096.

$$\frac{1000000}{729\times 4096}d$$

Multiply 729 and 4096 to get 2985984.

$$\frac{1000000}{2985984}d$$

Reduce the fraction \frac{1000000}{2985984} to lowest terms by extracting and canceling out 64.

$$\frac{15625}{46656}d$$

Differentiate w.r.t. d

$$\frac{15625}{46656} = 0.33489797668038407$$

Steps Using Definition of a Derivative

To raise a power to another power, multiply the exponents. Multiply 2 and 3 to get 6.

$$\frac{\mathrm{d}}{\mathrm{d}d}(\frac{4^{3}\times 25^{3}}{3^{6}\times \left(2^{3}\right)^{4}}d)$$

To raise a power to another power, multiply the exponents. Multiply 3 and 4 to get 12.

$$\frac{\mathrm{d}}{\mathrm{d}d}(\frac{4^{3}\times 25^{3}}{3^{6}\times 2^{12}}d)$$

Calculate 4 to the power of 3 and get 64.

$$\frac{\mathrm{d}}{\mathrm{d}d}(\frac{64\times 25^{3}}{3^{6}\times 2^{12}}d)$$

Calculate 25 to the power of 3 and get 15625.

$$\frac{\mathrm{d}}{\mathrm{d}d}(\frac{64\times 15625}{3^{6}\times 2^{12}}d)$$

Multiply 64 and 15625 to get 1000000.

$$\frac{\mathrm{d}}{\mathrm{d}d}(\frac{1000000}{3^{6}\times 2^{12}}d)$$

Calculate 3 to the power of 6 and get 729.

$$\frac{\mathrm{d}}{\mathrm{d}d}(\frac{1000000}{729\times 2^{12}}d)$$

Calculate 2 to the power of 12 and get 4096.

$$\frac{\mathrm{d}}{\mathrm{d}d}(\frac{1000000}{729\times 4096}d)$$

Multiply 729 and 4096 to get 2985984.

$$\frac{\mathrm{d}}{\mathrm{d}d}(\frac{1000000}{2985984}d)$$

Reduce the fraction \frac{1000000}{2985984} to lowest terms by extracting and canceling out 64.

$$\frac{\mathrm{d}}{\mathrm{d}d}(\frac{15625}{46656}d)$$

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

$$\frac{15625}{46656}d^{1-1}$$

Subtract 1 from 1.

$$\frac{15625}{46656}d^{0}$$

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

$$\frac{15625}{46656}\times 1$$

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

$$\frac{15625}{46656}$$