Calculate $75$ to the power of $2$ and get $5625$.
$$\sqrt{\frac{4745+5625+82^{2}+90^{2}}{5}-69}$$
Add $4745$ and $5625$ to get $10370$.
$$\sqrt{\frac{10370+82^{2}+90^{2}}{5}-69}$$
Calculate $82$ to the power of $2$ and get $6724$.
$$\sqrt{\frac{10370+6724+90^{2}}{5}-69}$$
Add $10370$ and $6724$ to get $17094$.
$$\sqrt{\frac{17094+90^{2}}{5}-69}$$
Calculate $90$ to the power of $2$ and get $8100$.
$$\sqrt{\frac{17094+8100}{5}-69}$$
Add $17094$ and $8100$ to get $25194$.
$$\sqrt{\frac{25194}{5}-69}$$
Convert $69$ to fraction $\frac{345}{5}$.
$$\sqrt{\frac{25194}{5}-\frac{345}{5}}$$
Since $\frac{25194}{5}$ and $\frac{345}{5}$ have the same denominator, subtract them by subtracting their numerators.
$$\sqrt{\frac{25194-345}{5}}$$
Subtract $345$ from $25194$ to get $24849$.
$$\sqrt{\frac{24849}{5}}$$
Rewrite the square root of the division $\sqrt{\frac{24849}{5}}$ as the division of square roots $\frac{\sqrt{24849}}{\sqrt{5}}$.
$$\frac{\sqrt{24849}}{\sqrt{5}}$$
Factor $24849=3^{2}\times 2761$. Rewrite the square root of the product $\sqrt{3^{2}\times 2761}$ as the product of square roots $\sqrt{3^{2}}\sqrt{2761}$. Take the square root of $3^{2}$.
$$\frac{3\sqrt{2761}}{\sqrt{5}}$$
Rationalize the denominator of $\frac{3\sqrt{2761}}{\sqrt{5}}$ by multiplying numerator and denominator by $\sqrt{5}$.
