Solve sqrt{23^2-2*23^2/90*13}

$$\sqrt { 23 ^ { 2 } - 2 \times \frac { 23 ^ { 2 } } { 90 } \times 13 }$$

Evaluate

$$\frac{92\sqrt{10}}{15}\approx 19.395302982$$

Solution Steps

Calculate 23 to the power of 2 and get 529.

$$\sqrt{529-2\times \frac{23^{2}}{90}\times 13}$$

Calculate 23 to the power of 2 and get 529.

$$\sqrt{529-2\times \frac{529}{90}\times 13}$$

Express 2\times \frac{529}{90} as a single fraction.

$$\sqrt{529-\frac{2\times 529}{90}\times 13}$$

Multiply 2 and 529 to get 1058.

$$\sqrt{529-\frac{1058}{90}\times 13}$$

Reduce the fraction \frac{1058}{90} to lowest terms by extracting and canceling out 2.

$$\sqrt{529-\frac{529}{45}\times 13}$$

Express \frac{529}{45}\times 13 as a single fraction.

$$\sqrt{529-\frac{529\times 13}{45}}$$

Multiply 529 and 13 to get 6877.

$$\sqrt{529-\frac{6877}{45}}$$

Convert 529 to fraction \frac{23805}{45}.

$$\sqrt{\frac{23805}{45}-\frac{6877}{45}}$$

Since \frac{23805}{45} and \frac{6877}{45} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{23805-6877}{45}}$$

Subtract 6877 from 23805 to get 16928.

$$\sqrt{\frac{16928}{45}}$$

Rewrite the square root of the division \sqrt{\frac{16928}{45}} as the division of square roots \frac{\sqrt{16928}}{\sqrt{45}}.

$$\frac{\sqrt{16928}}{\sqrt{45}}$$

Factor 16928=92^{2}\times 2. Rewrite the square root of the product \sqrt{92^{2}\times 2} as the product of square roots \sqrt{92^{2}}\sqrt{2}. Take the square root of 92^{2}.

$$\frac{92\sqrt{2}}{\sqrt{45}}$$

Factor 45=3^{2}\times 5. Rewrite the square root of the product \sqrt{3^{2}\times 5} as the product of square roots \sqrt{3^{2}}\sqrt{5}. Take the square root of 3^{2}.

$$\frac{92\sqrt{2}}{3\sqrt{5}}$$

Rationalize the denominator of \frac{92\sqrt{2}}{3\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{92\sqrt{2}\sqrt{5}}{3\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{92\sqrt{2}\sqrt{5}}{3\times 5}$$

To multiply \sqrt{2} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{92\sqrt{10}}{3\times 5}$$

Multiply 3 and 5 to get 15.

$$\frac{92\sqrt{10}}{15}$$