Solve (40*667)-(110*143)/sqrt{(40*350)-(110^2)sqrt{(40*549)-(143^2)}}

$$\frac{ (40 \times 667)-(110 \times 143) }{ \sqrt{ (40 \times 350)-( { 110 }^{ 2 } ) } \sqrt{ (40 \times 549)-( { 143 }^{ 2 } ) } }$$

Evaluate

$$\frac{1095\sqrt{28709}}{28709}\approx 6.462567415$$

Solution Steps

Multiply 40 and 667 to get 26680.

$$\frac{26680-110\times 143}{\sqrt{40\times 350-110^{2}}\sqrt{40\times 549-143^{2}}}$$

Multiply 110 and 143 to get 15730.

$$\frac{26680-15730}{\sqrt{40\times 350-110^{2}}\sqrt{40\times 549-143^{2}}}$$

Subtract 15730 from 26680 to get 10950.

$$\frac{10950}{\sqrt{40\times 350-110^{2}}\sqrt{40\times 549-143^{2}}}$$

Multiply 40 and 350 to get 14000.

$$\frac{10950}{\sqrt{14000-110^{2}}\sqrt{40\times 549-143^{2}}}$$

Calculate 110 to the power of 2 and get 12100.

$$\frac{10950}{\sqrt{14000-12100}\sqrt{40\times 549-143^{2}}}$$

Subtract 12100 from 14000 to get 1900.

$$\frac{10950}{\sqrt{1900}\sqrt{40\times 549-143^{2}}}$$

Factor 1900=10^{2}\times 19. Rewrite the square root of the product \sqrt{10^{2}\times 19} as the product of square roots \sqrt{10^{2}}\sqrt{19}. Take the square root of 10^{2}.

$$\frac{10950}{10\sqrt{19}\sqrt{40\times 549-143^{2}}}$$

Multiply 40 and 549 to get 21960.

$$\frac{10950}{10\sqrt{19}\sqrt{21960-143^{2}}}$$

Calculate 143 to the power of 2 and get 20449.

$$\frac{10950}{10\sqrt{19}\sqrt{21960-20449}}$$

Subtract 20449 from 21960 to get 1511.

$$\frac{10950}{10\sqrt{19}\sqrt{1511}}$$

To multiply \sqrt{19} and \sqrt{1511}, multiply the numbers under the square root.

$$\frac{10950}{10\sqrt{28709}}$$

Rationalize the denominator of \frac{10950}{10\sqrt{28709}} by multiplying numerator and denominator by \sqrt{28709}.

$$\frac{10950\sqrt{28709}}{10\left(\sqrt{28709}\right)^{2}}$$

The square of \sqrt{28709} is 28709.

$$\frac{10950\sqrt{28709}}{10\times 28709}$$

Cancel out 10 in both numerator and denominator.

$$\frac{1095\sqrt{28709}}{28709}$$