Solve sqrt{6.73*10^-11*5963*10^24/left(6371+640right)10^3}

$$\sqrt{ \frac{ 6.73 \times { 10 }^{ -11 } \times 5963 \times { 10 }^{ 24 } }{ \left( 6371+640 \right) { 10 }^{ 3 } } }$$

Evaluate

$$\frac{10000\sqrt{3126204121}}{2337}\approx 239248.901950628$$

Solution Steps

To multiply powers of the same base, add their exponents. Add -11 and 24 to get 13.

$$\sqrt{\frac{6.73\times 10^{13}\times 5963}{\left(6371+640\right)\times 10^{3}}}$$

Cancel out 10^{3} in both numerator and denominator.

$$\sqrt{\frac{6.73\times 5963\times 10^{10}}{640+6371}}$$

Multiply 6.73 and 5963 to get 40130.99.

$$\sqrt{\frac{40130.99\times 10^{10}}{640+6371}}$$

Calculate 10 to the power of 10 and get 10000000000.

$$\sqrt{\frac{40130.99\times 10000000000}{640+6371}}$$

Multiply 40130.99 and 10000000000 to get 401309900000000.

$$\sqrt{\frac{401309900000000}{640+6371}}$$

Add 640 and 6371 to get 7011.

$$\sqrt{\frac{401309900000000}{7011}}$$

Rewrite the square root of the division \sqrt{\frac{401309900000000}{7011}} as the division of square roots \frac{\sqrt{401309900000000}}{\sqrt{7011}}.

$$\frac{\sqrt{401309900000000}}{\sqrt{7011}}$$

Factor 401309900000000=10000^{2}\times 4013099. Rewrite the square root of the product \sqrt{10000^{2}\times 4013099} as the product of square roots \sqrt{10000^{2}}\sqrt{4013099}. Take the square root of 10000^{2}.

$$\frac{10000\sqrt{4013099}}{\sqrt{7011}}$$

Factor 7011=3^{2}\times 779. Rewrite the square root of the product \sqrt{3^{2}\times 779} as the product of square roots \sqrt{3^{2}}\sqrt{779}. Take the square root of 3^{2}.

$$\frac{10000\sqrt{4013099}}{3\sqrt{779}}$$

Rationalize the denominator of \frac{10000\sqrt{4013099}}{3\sqrt{779}} by multiplying numerator and denominator by \sqrt{779}.

$$\frac{10000\sqrt{4013099}\sqrt{779}}{3\left(\sqrt{779}\right)^{2}}$$

The square of \sqrt{779} is 779.

$$\frac{10000\sqrt{4013099}\sqrt{779}}{3\times 779}$$

To multiply \sqrt{4013099} and \sqrt{779}, multiply the numbers under the square root.

$$\frac{10000\sqrt{3126204121}}{3\times 779}$$

Multiply 3 and 779 to get 2337.

$$\frac{10000\sqrt{3126204121}}{2337}$$