Solve 5/frac{r^2{5^8}-5^2/8^2}

$$\frac { 5 } { \frac { r ^ { 2 } } { 5 ^ { 8 } } - \frac { 5 ^ { 2 } } { 8 ^ { 2 } } }$$

Evaluate

$$\frac{125000000}{64r^{2}-9765625}$$

Short Solution Steps

Calculate 5 to the power of 8 and get 390625.

$$\frac{5}{\frac{r^{2}}{390625}-\frac{5^{2}}{8^{2}}}$$

Calculate 5 to the power of 2 and get 25.

$$\frac{5}{\frac{r^{2}}{390625}-\frac{25}{8^{2}}}$$

Calculate 8 to the power of 2 and get 64.

$$\frac{5}{\frac{r^{2}}{390625}-\frac{25}{64}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 390625 and 64 is 25000000. Multiply \frac{r^{2}}{390625} times \frac{64}{64}. Multiply \frac{25}{64} times \frac{390625}{390625}.

$$\frac{5}{\frac{64r^{2}}{25000000}-\frac{25\times 390625}{25000000}}$$

Since \frac{64r^{2}}{25000000} and \frac{25\times 390625}{25000000} have the same denominator, subtract them by subtracting their numerators.

$$\frac{5}{\frac{64r^{2}-25\times 390625}{25000000}}$$

Do the multiplications in 64r^{2}-25\times 390625.

$$\frac{5}{\frac{64r^{2}-9765625}{25000000}}$$

Divide 5 by \frac{64r^{2}-9765625}{25000000} by multiplying 5 by the reciprocal of \frac{64r^{2}-9765625}{25000000}.

$$\frac{5\times 25000000}{64r^{2}-9765625}$$

Multiply 5 and 25000000 to get 125000000.

$$\frac{125000000}{64r^{2}-9765625}$$

Differentiate w.r.t. r

$$-\frac{16000000000r}{\left(64r^{2}-9765625\right)^{2}}$$