sqrt{20^2+60^2}+sqrt{20^2+40^2}+sqrt{{40}^{2}+{80}^{2}}+sqrt{{100}^{2}+{20}^{2}}+sqrt{{20}^{2}+{60}^{2}}+sqrt{{40}^2+{100}^2}+sqrt{{40}^2+{80}^2} eching

$$\sqrt{ { 20 }^{ 2 } + { 60 }^{ 2 } } + \sqrt{ { 20 }^{ 2 } + { 40 }^{ 2 } } + \sqrt{ { 40 }^{ 2 } + { 80 }^{ 2 } } + \sqrt{ { 100 }^{ 2 } + { 20 }^{ 2 } } + \sqrt{ { 20 }^{ 2 } + { 60 }^{ 2 } } + \sqrt{ { 40 }^{ 2 } + { 100 }^{ 2 } } + \sqrt{ { 40 }^{ 2 } + { 80 }^{ 2 } }$$

Baholash

$$20\left(\sqrt{26}+\sqrt{29}+2\sqrt{10}+5\sqrt{5}\right)\approx 559,781590571$$

Yechim qadamlari

2 daraja ko‘rsatkichini 20 ga hisoblang va 400 ni qiymatni oling.

$$\sqrt{400+60^{2}}+\sqrt{20^{2}+40^{2}}+\sqrt{40^{2}+80^{2}}+\sqrt{100^{2}+20^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{40^{2}+100^{2}}+\sqrt{40^{2}+80^{2}}$$

2 daraja ko‘rsatkichini 60 ga hisoblang va 3600 ni qiymatni oling.

$$\sqrt{400+3600}+\sqrt{20^{2}+40^{2}}+\sqrt{40^{2}+80^{2}}+\sqrt{100^{2}+20^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{40^{2}+100^{2}}+\sqrt{40^{2}+80^{2}}$$

4000 olish uchun 400 va 3600'ni qo'shing.

$$\sqrt{4000}+\sqrt{20^{2}+40^{2}}+\sqrt{40^{2}+80^{2}}+\sqrt{100^{2}+20^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{40^{2}+100^{2}}+\sqrt{40^{2}+80^{2}}$$

Faktor: 4000=20^{2}\times 10. \sqrt{20^{2}\times 10} koʻpaytmasining kvadrat ildizini \sqrt{20^{2}}\sqrt{10} kvadrat ildizlarining koʻpaytmasi sifatida qayta yozing. 20^{2} ning kvadrat ildizini chiqarish.

$$20\sqrt{10}+\sqrt{20^{2}+40^{2}}+\sqrt{40^{2}+80^{2}}+\sqrt{100^{2}+20^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{40^{2}+100^{2}}+\sqrt{40^{2}+80^{2}}$$

2 daraja ko‘rsatkichini 20 ga hisoblang va 400 ni qiymatni oling.

$$20\sqrt{10}+\sqrt{400+40^{2}}+\sqrt{40^{2}+80^{2}}+\sqrt{100^{2}+20^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{40^{2}+100^{2}}+\sqrt{40^{2}+80^{2}}$$

2 daraja ko‘rsatkichini 40 ga hisoblang va 1600 ni qiymatni oling.

$$20\sqrt{10}+\sqrt{400+1600}+\sqrt{40^{2}+80^{2}}+\sqrt{100^{2}+20^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{40^{2}+100^{2}}+\sqrt{40^{2}+80^{2}}$$

2000 olish uchun 400 va 1600'ni qo'shing.

$$20\sqrt{10}+\sqrt{2000}+\sqrt{40^{2}+80^{2}}+\sqrt{100^{2}+20^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{40^{2}+100^{2}}+\sqrt{40^{2}+80^{2}}$$

Faktor: 2000=20^{2}\times 5. \sqrt{20^{2}\times 5} koʻpaytmasining kvadrat ildizini \sqrt{20^{2}}\sqrt{5} kvadrat ildizlarining koʻpaytmasi sifatida qayta yozing. 20^{2} ning kvadrat ildizini chiqarish.

$$20\sqrt{10}+20\sqrt{5}+\sqrt{40^{2}+80^{2}}+\sqrt{100^{2}+20^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{40^{2}+100^{2}}+\sqrt{40^{2}+80^{2}}$$

2 daraja ko‘rsatkichini 40 ga hisoblang va 1600 ni qiymatni oling.

$$20\sqrt{10}+20\sqrt{5}+\sqrt{1600+80^{2}}+\sqrt{100^{2}+20^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{40^{2}+100^{2}}+\sqrt{40^{2}+80^{2}}$$

2 daraja ko‘rsatkichini 80 ga hisoblang va 6400 ni qiymatni oling.

$$20\sqrt{10}+20\sqrt{5}+\sqrt{1600+6400}+\sqrt{100^{2}+20^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{40^{2}+100^{2}}+\sqrt{40^{2}+80^{2}}$$

8000 olish uchun 1600 va 6400'ni qo'shing.

$$20\sqrt{10}+20\sqrt{5}+\sqrt{8000}+\sqrt{100^{2}+20^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{40^{2}+100^{2}}+\sqrt{40^{2}+80^{2}}$$

Faktor: 8000=40^{2}\times 5. \sqrt{40^{2}\times 5} koʻpaytmasining kvadrat ildizini \sqrt{40^{2}}\sqrt{5} kvadrat ildizlarining koʻpaytmasi sifatida qayta yozing. 40^{2} ning kvadrat ildizini chiqarish.

$$20\sqrt{10}+20\sqrt{5}+40\sqrt{5}+\sqrt{100^{2}+20^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{40^{2}+100^{2}}+\sqrt{40^{2}+80^{2}}$$

60\sqrt{5} ni olish uchun 20\sqrt{5} va 40\sqrt{5} ni birlashtirish.

$$20\sqrt{10}+60\sqrt{5}+\sqrt{100^{2}+20^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{40^{2}+100^{2}}+\sqrt{40^{2}+80^{2}}$$

2 daraja ko‘rsatkichini 100 ga hisoblang va 10000 ni qiymatni oling.

$$20\sqrt{10}+60\sqrt{5}+\sqrt{10000+20^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{40^{2}+100^{2}}+\sqrt{40^{2}+80^{2}}$$

2 daraja ko‘rsatkichini 20 ga hisoblang va 400 ni qiymatni oling.

$$20\sqrt{10}+60\sqrt{5}+\sqrt{10000+400}+\sqrt{20^{2}+60^{2}}+\sqrt{40^{2}+100^{2}}+\sqrt{40^{2}+80^{2}}$$

10400 olish uchun 10000 va 400'ni qo'shing.

$$20\sqrt{10}+60\sqrt{5}+\sqrt{10400}+\sqrt{20^{2}+60^{2}}+\sqrt{40^{2}+100^{2}}+\sqrt{40^{2}+80^{2}}$$

Faktor: 10400=20^{2}\times 26. \sqrt{20^{2}\times 26} koʻpaytmasining kvadrat ildizini \sqrt{20^{2}}\sqrt{26} kvadrat ildizlarining koʻpaytmasi sifatida qayta yozing. 20^{2} ning kvadrat ildizini chiqarish.

$$20\sqrt{10}+60\sqrt{5}+20\sqrt{26}+\sqrt{20^{2}+60^{2}}+\sqrt{40^{2}+100^{2}}+\sqrt{40^{2}+80^{2}}$$

2 daraja ko‘rsatkichini 20 ga hisoblang va 400 ni qiymatni oling.

$$20\sqrt{10}+60\sqrt{5}+20\sqrt{26}+\sqrt{400+60^{2}}+\sqrt{40^{2}+100^{2}}+\sqrt{40^{2}+80^{2}}$$

2 daraja ko‘rsatkichini 60 ga hisoblang va 3600 ni qiymatni oling.

$$20\sqrt{10}+60\sqrt{5}+20\sqrt{26}+\sqrt{400+3600}+\sqrt{40^{2}+100^{2}}+\sqrt{40^{2}+80^{2}}$$

4000 olish uchun 400 va 3600'ni qo'shing.

$$20\sqrt{10}+60\sqrt{5}+20\sqrt{26}+\sqrt{4000}+\sqrt{40^{2}+100^{2}}+\sqrt{40^{2}+80^{2}}$$

Faktor: 4000=20^{2}\times 10. \sqrt{20^{2}\times 10} koʻpaytmasining kvadrat ildizini \sqrt{20^{2}}\sqrt{10} kvadrat ildizlarining koʻpaytmasi sifatida qayta yozing. 20^{2} ning kvadrat ildizini chiqarish.

$$20\sqrt{10}+60\sqrt{5}+20\sqrt{26}+20\sqrt{10}+\sqrt{40^{2}+100^{2}}+\sqrt{40^{2}+80^{2}}$$

40\sqrt{10} ni olish uchun 20\sqrt{10} va 20\sqrt{10} ni birlashtirish.

$$40\sqrt{10}+60\sqrt{5}+20\sqrt{26}+\sqrt{40^{2}+100^{2}}+\sqrt{40^{2}+80^{2}}$$

2 daraja ko‘rsatkichini 40 ga hisoblang va 1600 ni qiymatni oling.

$$40\sqrt{10}+60\sqrt{5}+20\sqrt{26}+\sqrt{1600+100^{2}}+\sqrt{40^{2}+80^{2}}$$

2 daraja ko‘rsatkichini 100 ga hisoblang va 10000 ni qiymatni oling.

$$40\sqrt{10}+60\sqrt{5}+20\sqrt{26}+\sqrt{1600+10000}+\sqrt{40^{2}+80^{2}}$$

11600 olish uchun 1600 va 10000'ni qo'shing.

$$40\sqrt{10}+60\sqrt{5}+20\sqrt{26}+\sqrt{11600}+\sqrt{40^{2}+80^{2}}$$

Faktor: 11600=20^{2}\times 29. \sqrt{20^{2}\times 29} koʻpaytmasining kvadrat ildizini \sqrt{20^{2}}\sqrt{29} kvadrat ildizlarining koʻpaytmasi sifatida qayta yozing. 20^{2} ning kvadrat ildizini chiqarish.

$$40\sqrt{10}+60\sqrt{5}+20\sqrt{26}+20\sqrt{29}+\sqrt{40^{2}+80^{2}}$$

2 daraja ko‘rsatkichini 40 ga hisoblang va 1600 ni qiymatni oling.

$$40\sqrt{10}+60\sqrt{5}+20\sqrt{26}+20\sqrt{29}+\sqrt{1600+80^{2}}$$

2 daraja ko‘rsatkichini 80 ga hisoblang va 6400 ni qiymatni oling.

$$40\sqrt{10}+60\sqrt{5}+20\sqrt{26}+20\sqrt{29}+\sqrt{1600+6400}$$

8000 olish uchun 1600 va 6400'ni qo'shing.

$$40\sqrt{10}+60\sqrt{5}+20\sqrt{26}+20\sqrt{29}+\sqrt{8000}$$

Faktor: 8000=40^{2}\times 5. \sqrt{40^{2}\times 5} koʻpaytmasining kvadrat ildizini \sqrt{40^{2}}\sqrt{5} kvadrat ildizlarining koʻpaytmasi sifatida qayta yozing. 40^{2} ning kvadrat ildizini chiqarish.

$$40\sqrt{10}+60\sqrt{5}+20\sqrt{26}+20\sqrt{29}+40\sqrt{5}$$

100\sqrt{5} ni olish uchun 60\sqrt{5} va 40\sqrt{5} ni birlashtirish.

$$40\sqrt{10}+100\sqrt{5}+20\sqrt{26}+20\sqrt{29}$$

Omil

$$20 {(\sqrt{26} + \sqrt{29} + 2 \sqrt{10} + 5 \sqrt{5})} = 559,781590571$$