Solve 1^2/6^1+1^2/6^2-frac{{3}^{2}}{{6}^{3}}+frac{{4}^2}{{6}^4}-frac{{5}^2}{{6}^5}

$$\frac{ { 1 }^{ 2 } }{ { 6 }^{ 1 } } + \frac{ { 1 }^{ 2 } }{ { 6 }^{ 2 } } - \frac{ { 3 }^{ 2 } }{ { 6 }^{ 3 } } + \frac{ { 4 }^{ 2 } }{ { 6 }^{ 4 } } - \frac{ { 5 }^{ 2 } }{ { 6 }^{ 5 } }$$

Evaluate

$$\frac{1259}{7776}\approx 0.161908436$$

Solution Steps

Calculate 1 to the power of 2 and get 1.

$$\frac{1}{6^{1}}+\frac{1^{2}}{6^{2}}-\frac{3^{2}}{6^{3}}+\frac{4^{2}}{6^{4}}-\frac{5^{2}}{6^{5}}$$

Calculate 6 to the power of 1 and get 6.

$$\frac{1}{6}+\frac{1^{2}}{6^{2}}-\frac{3^{2}}{6^{3}}+\frac{4^{2}}{6^{4}}-\frac{5^{2}}{6^{5}}$$

Calculate 1 to the power of 2 and get 1.

$$\frac{1}{6}+\frac{1}{6^{2}}-\frac{3^{2}}{6^{3}}+\frac{4^{2}}{6^{4}}-\frac{5^{2}}{6^{5}}$$

Calculate 6 to the power of 2 and get 36.

$$\frac{1}{6}+\frac{1}{36}-\frac{3^{2}}{6^{3}}+\frac{4^{2}}{6^{4}}-\frac{5^{2}}{6^{5}}$$

Least common multiple of 6 and 36 is 36. Convert \frac{1}{6} and \frac{1}{36} to fractions with denominator 36.

$$\frac{6}{36}+\frac{1}{36}-\frac{3^{2}}{6^{3}}+\frac{4^{2}}{6^{4}}-\frac{5^{2}}{6^{5}}$$

Since \frac{6}{36} and \frac{1}{36} have the same denominator, add them by adding their numerators.

$$\frac{6+1}{36}-\frac{3^{2}}{6^{3}}+\frac{4^{2}}{6^{4}}-\frac{5^{2}}{6^{5}}$$

Add 6 and 1 to get 7.

$$\frac{7}{36}-\frac{3^{2}}{6^{3}}+\frac{4^{2}}{6^{4}}-\frac{5^{2}}{6^{5}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{7}{36}-\frac{9}{6^{3}}+\frac{4^{2}}{6^{4}}-\frac{5^{2}}{6^{5}}$$

Calculate 6 to the power of 3 and get 216.

$$\frac{7}{36}-\frac{9}{216}+\frac{4^{2}}{6^{4}}-\frac{5^{2}}{6^{5}}$$

Reduce the fraction \frac{9}{216} to lowest terms by extracting and canceling out 9.

$$\frac{7}{36}-\frac{1}{24}+\frac{4^{2}}{6^{4}}-\frac{5^{2}}{6^{5}}$$

Least common multiple of 36 and 24 is 72. Convert \frac{7}{36} and \frac{1}{24} to fractions with denominator 72.

$$\frac{14}{72}-\frac{3}{72}+\frac{4^{2}}{6^{4}}-\frac{5^{2}}{6^{5}}$$

Since \frac{14}{72} and \frac{3}{72} have the same denominator, subtract them by subtracting their numerators.

$$\frac{14-3}{72}+\frac{4^{2}}{6^{4}}-\frac{5^{2}}{6^{5}}$$

Subtract 3 from 14 to get 11.

$$\frac{11}{72}+\frac{4^{2}}{6^{4}}-\frac{5^{2}}{6^{5}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{11}{72}+\frac{16}{6^{4}}-\frac{5^{2}}{6^{5}}$$

Calculate 6 to the power of 4 and get 1296.

$$\frac{11}{72}+\frac{16}{1296}-\frac{5^{2}}{6^{5}}$$

Reduce the fraction \frac{16}{1296} to lowest terms by extracting and canceling out 16.

$$\frac{11}{72}+\frac{1}{81}-\frac{5^{2}}{6^{5}}$$

Least common multiple of 72 and 81 is 648. Convert \frac{11}{72} and \frac{1}{81} to fractions with denominator 648.

$$\frac{99}{648}+\frac{8}{648}-\frac{5^{2}}{6^{5}}$$

Since \frac{99}{648} and \frac{8}{648} have the same denominator, add them by adding their numerators.

$$\frac{99+8}{648}-\frac{5^{2}}{6^{5}}$$

Add 99 and 8 to get 107.

$$\frac{107}{648}-\frac{5^{2}}{6^{5}}$$

Calculate 5 to the power of 2 and get 25.

$$\frac{107}{648}-\frac{25}{6^{5}}$$

Calculate 6 to the power of 5 and get 7776.

$$\frac{107}{648}-\frac{25}{7776}$$

Least common multiple of 648 and 7776 is 7776. Convert \frac{107}{648} and \frac{25}{7776} to fractions with denominator 7776.

$$\frac{1284}{7776}-\frac{25}{7776}$$

Since \frac{1284}{7776} and \frac{25}{7776} have the same denominator, subtract them by subtracting their numerators.

$$\frac{1284-25}{7776}$$

Subtract 25 from 1284 to get 1259.

$$\frac{1259}{7776}$$

Factor

$$\frac{1259}{2 ^ {5} \cdot 3 ^ {5}} = 0.16190843621399176$$