Solve sqrt{218*37*2/95^143}

$$\sqrt { \frac { 218 \times 37 \cdot 2 } { 95 ^ { 143 } } }$$

Evaluate

$$\frac{2\sqrt{383135}}{24894280619189487224780035502836670501815837079606723166623624900521745091200861047541479044838686461062204902106032022857107222080230712890625}\approx 4.972861096 \cdot 10^{-140}$$

Solution Steps

Multiply 218 and 37 to get 8066.

$$\sqrt{\frac{8066\times 2}{95^{143}}}$$

Multiply 8066 and 2 to get 16132.

$$\sqrt{\frac{16132}{95^{143}}}$$

Calculate 95 to the power of 143 and get 6523423237336350731113846043380552965183182943784206469340426845587577021767813519163099894013076194246147452494368995418749528183919349488656221391567982995551409875783892935163981599206793508183555226422261238060004843792597177027861119077110918507145242983824573457241058349609375.

$$\sqrt{\frac{16132}{6523423237336350731113846043380552965183182943784206469340426845587577021767813519163099894013076194246147452494368995418749528183919349488656221391567982995551409875783892935163981599206793508183555226422261238060004843792597177027861119077110918507145242983824573457241058349609375}}$$

Rewrite the square root of the division \sqrt{\frac{16132}{6523423237336350731113846043380552965183182943784206469340426845587577021767813519163099894013076194246147452494368995418749528183919349488656221391567982995551409875783892935163981599206793508183555226422261238060004843792597177027861119077110918507145242983824573457241058349609375}} as the division of square roots \frac{\sqrt{16132}}{\sqrt{6523423237336350731113846043380552965183182943784206469340426845587577021767813519163099894013076194246147452494368995418749528183919349488656221391567982995551409875783892935163981599206793508183555226422261238060004843792597177027861119077110918507145242983824573457241058349609375}}.

$$\frac{\sqrt{16132}}{\sqrt{6523423237336350731113846043380552965183182943784206469340426845587577021767813519163099894013076194246147452494368995418749528183919349488656221391567982995551409875783892935163981599206793508183555226422261238060004843792597177027861119077110918507145242983824573457241058349609375}}$$

Factor 16132=2^{2}\times 4033. Rewrite the square root of the product \sqrt{2^{2}\times 4033} as the product of square roots \sqrt{2^{2}}\sqrt{4033}. Take the square root of 2^{2}.

$$\frac{2\sqrt{4033}}{\sqrt{6523423237336350731113846043380552965183182943784206469340426845587577021767813519163099894013076194246147452494368995418749528183919349488656221391567982995551409875783892935163981599206793508183555226422261238060004843792597177027861119077110918507145242983824573457241058349609375}}$$

Factor 6523423237336350731113846043380552965183182943784206469340426845587577021767813519163099894013076194246147452494368995418749528183919349488656221391567982995551409875783892935163981599206793508183555226422261238060004843792597177027861119077110918507145242983824573457241058349609375=262045059149363023418737215819333373703324600837965507017090788426544685170535379447805042577249331169075841074800337082706391811370849609375^{2}\times 95. Rewrite the square root of the product \sqrt{262045059149363023418737215819333373703324600837965507017090788426544685170535379447805042577249331169075841074800337082706391811370849609375^{2}\times 95} as the product of square roots \sqrt{262045059149363023418737215819333373703324600837965507017090788426544685170535379447805042577249331169075841074800337082706391811370849609375^{2}}\sqrt{95}. Take the square root of 262045059149363023418737215819333373703324600837965507017090788426544685170535379447805042577249331169075841074800337082706391811370849609375^{2}.

$$\frac{2\sqrt{4033}}{262045059149363023418737215819333373703324600837965507017090788426544685170535379447805042577249331169075841074800337082706391811370849609375\sqrt{95}}$$

Rationalize the denominator of \frac{2\sqrt{4033}}{262045059149363023418737215819333373703324600837965507017090788426544685170535379447805042577249331169075841074800337082706391811370849609375\sqrt{95}} by multiplying numerator and denominator by \sqrt{95}.

$$\frac{2\sqrt{4033}\sqrt{95}}{262045059149363023418737215819333373703324600837965507017090788426544685170535379447805042577249331169075841074800337082706391811370849609375\left(\sqrt{95}\right)^{2}}$$

The square of \sqrt{95} is 95.

$$\frac{2\sqrt{4033}\sqrt{95}}{262045059149363023418737215819333373703324600837965507017090788426544685170535379447805042577249331169075841074800337082706391811370849609375\times 95}$$

To multiply \sqrt{4033} and \sqrt{95}, multiply the numbers under the square root.

$$\frac{2\sqrt{383135}}{262045059149363023418737215819333373703324600837965507017090788426544685170535379447805042577249331169075841074800337082706391811370849609375\times 95}$$

Multiply 262045059149363023418737215819333373703324600837965507017090788426544685170535379447805042577249331169075841074800337082706391811370849609375 and 95 to get 24894280619189487224780035502836670501815837079606723166623624900521745091200861047541479044838686461062204902106032022857107222080230712890625.

$$\frac{2\sqrt{383135}}{24894280619189487224780035502836670501815837079606723166623624900521745091200861047541479044838686461062204902106032022857107222080230712890625}$$