Solve sqrt{10^4}-sqrt{10^3}+sqrt{10^-2}-sqrt{10^-3}

$$\sqrt{ { 10 }^{ 4 } } - \sqrt{ { 10 }^{ 3 } } + \sqrt{ { 10 }^{ -2 } } - \sqrt{ { 10 }^{ -3 } }$$

Evaluate

$$-\frac{1001\sqrt{10}}{100}+\frac{1001}{10}\approx 68.445600622$$

Short Solution Steps

Calculate 10 to the power of 4 and get 10000.

$$\sqrt{10000}-\sqrt{10^{3}}+\sqrt{10^{-2}}-\sqrt{10^{-3}}$$

Calculate the square root of 10000 and get 100.

$$100-\sqrt{10^{3}}+\sqrt{10^{-2}}-\sqrt{10^{-3}}$$

Calculate 10 to the power of 3 and get 1000.

$$100-\sqrt{1000}+\sqrt{10^{-2}}-\sqrt{10^{-3}}$$

Factor 1000=10^{2}\times 10. Rewrite the square root of the product \sqrt{10^{2}\times 10} as the product of square roots \sqrt{10^{2}}\sqrt{10}. Take the square root of 10^{2}.

$$100-10\sqrt{10}+\sqrt{10^{-2}}-\sqrt{10^{-3}}$$

Calculate 10 to the power of -2 and get \frac{1}{100}.

$$100-10\sqrt{10}+\sqrt{\frac{1}{100}}-\sqrt{10^{-3}}$$

Rewrite the square root of the division \frac{1}{100} as the division of square roots \frac{\sqrt{1}}{\sqrt{100}}. Take the square root of both numerator and denominator.

$$100-10\sqrt{10}+\frac{1}{10}-\sqrt{10^{-3}}$$

Add 100 and \frac{1}{10} to get \frac{1001}{10}.

$$\frac{1001}{10}-10\sqrt{10}-\sqrt{10^{-3}}$$

Calculate 10 to the power of -3 and get \frac{1}{1000}.

$$\frac{1001}{10}-10\sqrt{10}-\sqrt{\frac{1}{1000}}$$

Rewrite the square root of the division \sqrt{\frac{1}{1000}} as the division of square roots \frac{\sqrt{1}}{\sqrt{1000}}.

$$\frac{1001}{10}-10\sqrt{10}-\frac{\sqrt{1}}{\sqrt{1000}}$$

Calculate the square root of 1 and get 1.

$$\frac{1001}{10}-10\sqrt{10}-\frac{1}{\sqrt{1000}}$$

Factor 1000=10^{2}\times 10. Rewrite the square root of the product \sqrt{10^{2}\times 10} as the product of square roots \sqrt{10^{2}}\sqrt{10}. Take the square root of 10^{2}.

$$\frac{1001}{10}-10\sqrt{10}-\frac{1}{10\sqrt{10}}$$

Rationalize the denominator of \frac{1}{10\sqrt{10}} by multiplying numerator and denominator by \sqrt{10}.

$$\frac{1001}{10}-10\sqrt{10}-\frac{\sqrt{10}}{10\left(\sqrt{10}\right)^{2}}$$

The square of \sqrt{10} is 10.

$$\frac{1001}{10}-10\sqrt{10}-\frac{\sqrt{10}}{10\times 10}$$

Multiply 10 and 10 to get 100.

$$\frac{1001}{10}-10\sqrt{10}-\frac{\sqrt{10}}{100}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 10 and 100 is 100. Multiply \frac{1001}{10} times \frac{10}{10}.

$$\frac{1001\times 10}{100}-10\sqrt{10}-\frac{\sqrt{10}}{100}$$

Since \frac{1001\times 10}{100} and \frac{\sqrt{10}}{100} have the same denominator, subtract them by subtracting their numerators.

$$\frac{1001\times 10-\sqrt{10}}{100}-10\sqrt{10}$$

Do the multiplications in 1001\times 10-\sqrt{10}.

$$\frac{10010-\sqrt{10}}{100}-10\sqrt{10}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -10\sqrt{10} times \frac{100}{100}.

$$\frac{10010-\sqrt{10}}{100}+\frac{100\left(-10\right)\sqrt{10}}{100}$$

Since \frac{10010-\sqrt{10}}{100} and \frac{100\left(-10\right)\sqrt{10}}{100} have the same denominator, add them by adding their numerators.

$$\frac{10010-\sqrt{10}+100\left(-10\right)\sqrt{10}}{100}$$

Do the multiplications in 10010-\sqrt{10}+100\left(-10\right)\sqrt{10}.

$$\frac{10010-\sqrt{10}-1000\sqrt{10}}{100}$$

Do the calculations in 10010-\sqrt{10}-1000\sqrt{10}.

$$\frac{10010-1001\sqrt{10}}{100}$$