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

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

Evaluate

$$\frac{5}{7}\approx 0.714285714$$

Solution Steps

Calculate 1 to the power of 3 and get 1.

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

Calculate 2 to the power of 3 and get 8.

$$\sqrt{\frac{1+8+3^{3}+4^{3}+5^{3}}{1^{3}+2^{3}+3^{3}+4^{3}+5^{3}+6^{3}}}$$

Add 1 and 8 to get 9.

$$\sqrt{\frac{9+3^{3}+4^{3}+5^{3}}{1^{3}+2^{3}+3^{3}+4^{3}+5^{3}+6^{3}}}$$

Calculate 3 to the power of 3 and get 27.

$$\sqrt{\frac{9+27+4^{3}+5^{3}}{1^{3}+2^{3}+3^{3}+4^{3}+5^{3}+6^{3}}}$$

Add 9 and 27 to get 36.

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

Calculate 4 to the power of 3 and get 64.

$$\sqrt{\frac{36+64+5^{3}}{1^{3}+2^{3}+3^{3}+4^{3}+5^{3}+6^{3}}}$$

Add 36 and 64 to get 100.

$$\sqrt{\frac{100+5^{3}}{1^{3}+2^{3}+3^{3}+4^{3}+5^{3}+6^{3}}}$$

Calculate 5 to the power of 3 and get 125.

$$\sqrt{\frac{100+125}{1^{3}+2^{3}+3^{3}+4^{3}+5^{3}+6^{3}}}$$

Add 100 and 125 to get 225.

$$\sqrt{\frac{225}{1^{3}+2^{3}+3^{3}+4^{3}+5^{3}+6^{3}}}$$

Calculate 1 to the power of 3 and get 1.

$$\sqrt{\frac{225}{1+2^{3}+3^{3}+4^{3}+5^{3}+6^{3}}}$$

Calculate 2 to the power of 3 and get 8.

$$\sqrt{\frac{225}{1+8+3^{3}+4^{3}+5^{3}+6^{3}}}$$

Add 1 and 8 to get 9.

$$\sqrt{\frac{225}{9+3^{3}+4^{3}+5^{3}+6^{3}}}$$

Calculate 3 to the power of 3 and get 27.

$$\sqrt{\frac{225}{9+27+4^{3}+5^{3}+6^{3}}}$$

Add 9 and 27 to get 36.

$$\sqrt{\frac{225}{36+4^{3}+5^{3}+6^{3}}}$$

Calculate 4 to the power of 3 and get 64.

$$\sqrt{\frac{225}{36+64+5^{3}+6^{3}}}$$

Add 36 and 64 to get 100.

$$\sqrt{\frac{225}{100+5^{3}+6^{3}}}$$

Calculate 5 to the power of 3 and get 125.

$$\sqrt{\frac{225}{100+125+6^{3}}}$$

Add 100 and 125 to get 225.

$$\sqrt{\frac{225}{225+6^{3}}}$$

Calculate 6 to the power of 3 and get 216.

$$\sqrt{\frac{225}{225+216}}$$

Add 225 and 216 to get 441.

$$\sqrt{\frac{225}{441}}$$

Reduce the fraction \frac{225}{441} to lowest terms by extracting and canceling out 9.

$$\sqrt{\frac{25}{49}}$$

Rewrite the square root of the division \frac{25}{49} as the division of square roots \frac{\sqrt{25}}{\sqrt{49}}. Take the square root of both numerator and denominator.

$$\frac{5}{7}$$

Factor

$$\frac{5}{7} = 0.7142857142857143$$