Solve O(7/3)^5*(8/21)^0div(7/4)^4

$$O ( \frac { 7 } { 3 } ) ^ { 5 } \times ( \frac { 8 } { 21 } ) ^ { 0 } \div ( \frac { 7 } { 4 } ) ^ { 4 }$$

Evaluate

$$\frac{1792O}{243}$$

Solution Steps

Calculate \frac{7}{3} to the power of 5 and get \frac{16807}{243}.

$$\frac{O\times \frac{16807}{243}\times \left(\frac{8}{21}\right)^{0}}{\left(\frac{7}{4}\right)^{4}}$$

Calculate \frac{8}{21} to the power of 0 and get 1.

$$\frac{O\times \frac{16807}{243}\times 1}{\left(\frac{7}{4}\right)^{4}}$$

Multiply \frac{16807}{243} and 1 to get \frac{16807}{243}.

$$\frac{O\times \frac{16807}{243}}{\left(\frac{7}{4}\right)^{4}}$$

Calculate \frac{7}{4} to the power of 4 and get \frac{2401}{256}.

$$\frac{O\times \frac{16807}{243}}{\frac{2401}{256}}$$

Divide O\times \frac{16807}{243} by \frac{2401}{256} by multiplying O\times \frac{16807}{243} by the reciprocal of \frac{2401}{256}.

$$\frac{O\times \frac{16807}{243}\times 256}{2401}$$

Multiply \frac{16807}{243} and 256 to get \frac{4302592}{243}.

$$\frac{O\times \frac{4302592}{243}}{2401}$$

Divide O\times \frac{4302592}{243} by 2401 to get O\times \frac{1792}{243}.

$$O\times \frac{1792}{243}$$

Differentiate w.r.t. O

$$\frac{1792}{243} = 7\frac{91}{243} = 7.374485596707819$$

Steps Using Definition of a Derivative

Calculate \frac{7}{3} to the power of 5 and get \frac{16807}{243}.

$$\frac{\mathrm{d}}{\mathrm{d}O}(\frac{O\times \frac{16807}{243}\times \left(\frac{8}{21}\right)^{0}}{\left(\frac{7}{4}\right)^{4}})$$

Calculate \frac{8}{21} to the power of 0 and get 1.

$$\frac{\mathrm{d}}{\mathrm{d}O}(\frac{O\times \frac{16807}{243}\times 1}{\left(\frac{7}{4}\right)^{4}})$$

Multiply \frac{16807}{243} and 1 to get \frac{16807}{243}.

$$\frac{\mathrm{d}}{\mathrm{d}O}(\frac{O\times \frac{16807}{243}}{\left(\frac{7}{4}\right)^{4}})$$

Calculate \frac{7}{4} to the power of 4 and get \frac{2401}{256}.

$$\frac{\mathrm{d}}{\mathrm{d}O}(\frac{O\times \frac{16807}{243}}{\frac{2401}{256}})$$

Divide O\times \frac{16807}{243} by \frac{2401}{256} by multiplying O\times \frac{16807}{243} by the reciprocal of \frac{2401}{256}.

$$\frac{\mathrm{d}}{\mathrm{d}O}(\frac{O\times \frac{16807}{243}\times 256}{2401})$$

Multiply \frac{16807}{243} and 256 to get \frac{4302592}{243}.

$$\frac{\mathrm{d}}{\mathrm{d}O}(\frac{O\times \frac{4302592}{243}}{2401})$$

Divide O\times \frac{4302592}{243} by 2401 to get O\times \frac{1792}{243}.

$$\frac{\mathrm{d}}{\mathrm{d}O}(O\times \frac{1792}{243})$$

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

$$\frac{1792}{243}O^{1-1}$$

Subtract 1 from 1.

$$\frac{1792}{243}O^{0}$$

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

$$\frac{1792}{243}\times 1$$

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

$$\frac{1792}{243}$$