Solve 3^0/3^1*3^2/3^3*frac{{3}^{4}}{{3}^{5}}*frac{{3}^{6}}{{3}^{7}}*frac{{3}^8}{{3}^9}*frac{{3}^10}{{3}^11}

$$\frac{ { 3 }^{ 0 } }{ { 3 }^{ 1 } } \times \frac{ { 3 }^{ 2 } }{ { 3 }^{ 3 } } \times \frac{ { 3 }^{ 4 } }{ { 3 }^{ 5 } } \times \frac{ { 3 }^{ 6 } }{ { 3 }^{ 7 } } \times \frac{ { 3 }^{ 8 } }{ { 3 }^{ 9 } } \times \frac{ { 3 }^{ 10 } }{ { 3 }^{ 11 } }$$

Evaluate

$$\frac{1}{729}\approx 0.001371742$$

Solution Steps

Rewrite 3^{1} as 3\times 3^{0}. Cancel out 3^{0} in both numerator and denominator.

$$\frac{1}{3}\times \frac{3^{2}}{3^{3}}\times \frac{3^{4}}{3^{5}}\times \frac{3^{6}}{3^{7}}\times \frac{3^{8}}{3^{9}}\times \frac{3^{10}}{3^{11}}$$

Rewrite 3^{3} as 3\times 3^{2}. Cancel out 3^{2} in both numerator and denominator.

$$\frac{1}{3}\times \frac{1}{3}\times \frac{3^{4}}{3^{5}}\times \frac{3^{6}}{3^{7}}\times \frac{3^{8}}{3^{9}}\times \frac{3^{10}}{3^{11}}$$

Rewrite 3^{5} as 3\times 3^{4}. Cancel out 3^{4} in both numerator and denominator.

$$\frac{1}{3}\times \frac{1}{3}\times \frac{1}{3}\times \frac{3^{6}}{3^{7}}\times \frac{3^{8}}{3^{9}}\times \frac{3^{10}}{3^{11}}$$

Rewrite 3^{7} as 3\times 3^{6}. Cancel out 3^{6} in both numerator and denominator.

$$\frac{1}{3}\times \frac{1}{3}\times \frac{1}{3}\times \frac{1}{3}\times \frac{3^{8}}{3^{9}}\times \frac{3^{10}}{3^{11}}$$

Rewrite 3^{9} as 3\times 3^{8}. Cancel out 3^{8} in both numerator and denominator.

$$\frac{1}{3}\times \frac{1}{3}\times \frac{1}{3}\times \frac{1}{3}\times \frac{1}{3}\times \frac{3^{10}}{3^{11}}$$

Rewrite 3^{11} as 3\times 3^{10}. Cancel out 3^{10} in both numerator and denominator.

$$\frac{1}{3}\times \frac{1}{3}\times \frac{1}{3}\times \frac{1}{3}\times \frac{1}{3}\times \frac{1}{3}$$

Multiply \frac{1}{3} and \frac{1}{3} to get \left(\frac{1}{3}\right)^{2}.

$$\left(\frac{1}{3}\right)^{2}\times \frac{1}{3}\times \frac{1}{3}\times \frac{1}{3}\times \frac{1}{3}$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$\left(\frac{1}{3}\right)^{3}\times \frac{1}{3}\times \frac{1}{3}\times \frac{1}{3}$$

To multiply powers of the same base, add their exponents. Add 3 and 1 to get 4.

$$\left(\frac{1}{3}\right)^{4}\times \frac{1}{3}\times \frac{1}{3}$$

To multiply powers of the same base, add their exponents. Add 4 and 1 to get 5.

$$\left(\frac{1}{3}\right)^{5}\times \frac{1}{3}$$

To multiply powers of the same base, add their exponents. Add 5 and 1 to get 6.

$$\left(\frac{1}{3}\right)^{6}$$

Calculate \frac{1}{3} to the power of 6 and get \frac{1}{729}.

$$\frac{1}{729}$$

Factor

$$\frac{1}{3 ^ {6}} = 0.0013717421124828531$$