Solve frac{sqrt{126}*sqrt{63}*sqrt{45}}{sqrt{147}*sqrt{243}}

$$\frac { \sqrt { 126 } \times \sqrt { 63 } \times \sqrt { 45 } } { \sqrt { 147 } \times \sqrt { 243 } }$$

Evaluate

$$\sqrt{10}\approx 3.16227766$$

Solution Steps

Factor 126=63\times 2. Rewrite the square root of the product \sqrt{63\times 2} as the product of square roots \sqrt{63}\sqrt{2}.

$$\frac{\sqrt{63}\sqrt{2}\sqrt{63}\sqrt{45}}{\sqrt{147}\sqrt{243}}$$

Multiply \sqrt{63} and \sqrt{63} to get 63.

$$\frac{63\sqrt{2}\sqrt{45}}{\sqrt{147}\sqrt{243}}$$

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}.

$$\frac{63\sqrt{2}\times 3\sqrt{5}}{\sqrt{147}\sqrt{243}}$$

Multiply 63 and 3 to get 189.

$$\frac{189\sqrt{2}\sqrt{5}}{\sqrt{147}\sqrt{243}}$$

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

$$\frac{189\sqrt{10}}{\sqrt{147}\sqrt{243}}$$

Factor 147=7^{2}\times 3. Rewrite the square root of the product \sqrt{7^{2}\times 3} as the product of square roots \sqrt{7^{2}}\sqrt{3}. Take the square root of 7^{2}.

$$\frac{189\sqrt{10}}{7\sqrt{3}\sqrt{243}}$$

Factor 243=9^{2}\times 3. Rewrite the square root of the product \sqrt{9^{2}\times 3} as the product of square roots \sqrt{9^{2}}\sqrt{3}. Take the square root of 9^{2}.

$$\frac{189\sqrt{10}}{7\sqrt{3}\times 9\sqrt{3}}$$

Multiply 7 and 9 to get 63.

$$\frac{189\sqrt{10}}{63\sqrt{3}\sqrt{3}}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$\frac{189\sqrt{10}}{63\times 3}$$

Multiply 63 and 3 to get 189.

$$\frac{189\sqrt{10}}{189}$$

Cancel out 189 and 189.

$$\sqrt{10}$$