Solve frac{12^{3}*(m^{4})^{2}*4^16}{(m^3)^2*144*2^32}

$$\frac { 12 ^ { 3 } \times ( m ^ { 4 } ) ^ { 2 } \times 4 ^ { 16 } } { ( m ^ { 3 } ) ^ { 2 } \times 144 \times 2 ^ { 32 } }$$

Evaluate

$$12m^{2}$$

Solution Steps

To raise a power to another power, multiply the exponents. Multiply 4 and 2 to get 8.

$$\frac{12^{3}m^{8}\times 4^{16}}{\left(m^{3}\right)^{2}\times 144\times 2^{32}}$$

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

$$\frac{12^{3}m^{8}\times 4^{16}}{m^{6}\times 144\times 2^{32}}$$

Cancel out m^{6} in both numerator and denominator.

$$\frac{4^{16}\times 12^{3}m^{2}}{144\times 2^{32}}$$

Calculate 4 to the power of 16 and get 4294967296.

$$\frac{4294967296\times 12^{3}m^{2}}{144\times 2^{32}}$$

Calculate 12 to the power of 3 and get 1728.

$$\frac{4294967296\times 1728m^{2}}{144\times 2^{32}}$$

Multiply 4294967296 and 1728 to get 7421703487488.

$$\frac{7421703487488m^{2}}{144\times 2^{32}}$$

Calculate 2 to the power of 32 and get 4294967296.

$$\frac{7421703487488m^{2}}{144\times 4294967296}$$

Multiply 144 and 4294967296 to get 618475290624.

$$\frac{7421703487488m^{2}}{618475290624}$$

Divide 7421703487488m^{2} by 618475290624 to get 12m^{2}.

$$12m^{2}$$

Differentiate w.r.t. m

$$24m$$

Steps Using Definition of a Derivative

To raise a power to another power, multiply the exponents. Multiply 4 and 2 to get 8.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{12^{3}m^{8}\times 4^{16}}{\left(m^{3}\right)^{2}\times 144\times 2^{32}})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{12^{3}m^{8}\times 4^{16}}{m^{6}\times 144\times 2^{32}})$$

Cancel out m^{6} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{4^{16}\times 12^{3}m^{2}}{144\times 2^{32}})$$

Calculate 4 to the power of 16 and get 4294967296.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{4294967296\times 12^{3}m^{2}}{144\times 2^{32}})$$

Calculate 12 to the power of 3 and get 1728.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{4294967296\times 1728m^{2}}{144\times 2^{32}})$$

Multiply 4294967296 and 1728 to get 7421703487488.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{7421703487488m^{2}}{144\times 2^{32}})$$

Calculate 2 to the power of 32 and get 4294967296.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{7421703487488m^{2}}{144\times 4294967296})$$

Multiply 144 and 4294967296 to get 618475290624.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{7421703487488m^{2}}{618475290624})$$

Divide 7421703487488m^{2} by 618475290624 to get 12m^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}m}(12m^{2})$$

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

$$2\times 12m^{2-1}$$

Multiply 2 times 12.

$$24m^{2-1}$$

Subtract 1 from 2.

$$24m^{1}$$

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

$$24m$$