185*sqrt{20^2+60^2}+185*sqrt{20^2+40^2}+sqrt{{20}^{2}+{40}^{2}}+sqrt{{40}^{2}+{80}^{2}}+sqrt{{20}^{2}+{60}^{2}}+sqrt{{20}^2+{80}^2}+sqrt{{40}^2+{100}^2}= eching

$$185 \times \sqrt{ { 20 }^{ 2 } + { 60 }^{ 2 } } +185 \times \sqrt{ { 20 }^{ 2 } + { 40 }^{ 2 } } + \sqrt{ { 20 }^{ 2 } + { 40 }^{ 2 } } + \sqrt{ { 40 }^{ 2 } + { 80 }^{ 2 } } + \sqrt{ { 20 }^{ 2 } + { 60 }^{ 2 } } + \sqrt{ { 20 }^{ 2 } + { 80 }^{ 2 } } + \sqrt{ { 40 }^{ 2 } + { 100 }^{ 2 } } =$$

Baholash

$$20\left(\sqrt{17}+\sqrt{29}+186\sqrt{10}+188\sqrt{5}\right)\approx 20361,453899881$$

Yechim qadamlari

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

$$185\sqrt{400+60^{2}}+185\sqrt{20^{2}+40^{2}}+\sqrt{20^{2}+40^{2}}+\sqrt{40^{2}+80^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{20^{2}+80^{2}}+\sqrt{40^{2}+100^{2}}$$

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

$$185\sqrt{400+3600}+185\sqrt{20^{2}+40^{2}}+\sqrt{20^{2}+40^{2}}+\sqrt{40^{2}+80^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{20^{2}+80^{2}}+\sqrt{40^{2}+100^{2}}$$

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

$$185\sqrt{4000}+185\sqrt{20^{2}+40^{2}}+\sqrt{20^{2}+40^{2}}+\sqrt{40^{2}+80^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{20^{2}+80^{2}}+\sqrt{40^{2}+100^{2}}$$

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.

$$185\times 20\sqrt{10}+185\sqrt{20^{2}+40^{2}}+\sqrt{20^{2}+40^{2}}+\sqrt{40^{2}+80^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{20^{2}+80^{2}}+\sqrt{40^{2}+100^{2}}$$

3700 hosil qilish uchun 185 va 20 ni ko'paytirish.

$$3700\sqrt{10}+185\sqrt{20^{2}+40^{2}}+\sqrt{20^{2}+40^{2}}+\sqrt{40^{2}+80^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{20^{2}+80^{2}}+\sqrt{40^{2}+100^{2}}$$

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

$$3700\sqrt{10}+185\sqrt{400+40^{2}}+\sqrt{20^{2}+40^{2}}+\sqrt{40^{2}+80^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{20^{2}+80^{2}}+\sqrt{40^{2}+100^{2}}$$

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

$$3700\sqrt{10}+185\sqrt{400+1600}+\sqrt{20^{2}+40^{2}}+\sqrt{40^{2}+80^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{20^{2}+80^{2}}+\sqrt{40^{2}+100^{2}}$$

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

$$3700\sqrt{10}+185\sqrt{2000}+\sqrt{20^{2}+40^{2}}+\sqrt{40^{2}+80^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{20^{2}+80^{2}}+\sqrt{40^{2}+100^{2}}$$

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.

$$3700\sqrt{10}+185\times 20\sqrt{5}+\sqrt{20^{2}+40^{2}}+\sqrt{40^{2}+80^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{20^{2}+80^{2}}+\sqrt{40^{2}+100^{2}}$$

3700 hosil qilish uchun 185 va 20 ni ko'paytirish.

$$3700\sqrt{10}+3700\sqrt{5}+\sqrt{20^{2}+40^{2}}+\sqrt{40^{2}+80^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{20^{2}+80^{2}}+\sqrt{40^{2}+100^{2}}$$

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

$$3700\sqrt{10}+3700\sqrt{5}+\sqrt{400+40^{2}}+\sqrt{40^{2}+80^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{20^{2}+80^{2}}+\sqrt{40^{2}+100^{2}}$$

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

$$3700\sqrt{10}+3700\sqrt{5}+\sqrt{400+1600}+\sqrt{40^{2}+80^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{20^{2}+80^{2}}+\sqrt{40^{2}+100^{2}}$$

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

$$3700\sqrt{10}+3700\sqrt{5}+\sqrt{2000}+\sqrt{40^{2}+80^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{20^{2}+80^{2}}+\sqrt{40^{2}+100^{2}}$$

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.

$$3700\sqrt{10}+3700\sqrt{5}+20\sqrt{5}+\sqrt{40^{2}+80^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{20^{2}+80^{2}}+\sqrt{40^{2}+100^{2}}$$

3720\sqrt{5} ni olish uchun 3700\sqrt{5} va 20\sqrt{5} ni birlashtirish.

$$3700\sqrt{10}+3720\sqrt{5}+\sqrt{40^{2}+80^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{20^{2}+80^{2}}+\sqrt{40^{2}+100^{2}}$$

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

$$3700\sqrt{10}+3720\sqrt{5}+\sqrt{1600+80^{2}}+\sqrt{20^{2}+60^{2}}+\sqrt{20^{2}+80^{2}}+\sqrt{40^{2}+100^{2}}$$

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

$$3700\sqrt{10}+3720\sqrt{5}+\sqrt{1600+6400}+\sqrt{20^{2}+60^{2}}+\sqrt{20^{2}+80^{2}}+\sqrt{40^{2}+100^{2}}$$

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

$$3700\sqrt{10}+3720\sqrt{5}+\sqrt{8000}+\sqrt{20^{2}+60^{2}}+\sqrt{20^{2}+80^{2}}+\sqrt{40^{2}+100^{2}}$$

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.

$$3700\sqrt{10}+3720\sqrt{5}+40\sqrt{5}+\sqrt{20^{2}+60^{2}}+\sqrt{20^{2}+80^{2}}+\sqrt{40^{2}+100^{2}}$$

3760\sqrt{5} ni olish uchun 3720\sqrt{5} va 40\sqrt{5} ni birlashtirish.

$$3700\sqrt{10}+3760\sqrt{5}+\sqrt{20^{2}+60^{2}}+\sqrt{20^{2}+80^{2}}+\sqrt{40^{2}+100^{2}}$$

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

$$3700\sqrt{10}+3760\sqrt{5}+\sqrt{400+60^{2}}+\sqrt{20^{2}+80^{2}}+\sqrt{40^{2}+100^{2}}$$

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

$$3700\sqrt{10}+3760\sqrt{5}+\sqrt{400+3600}+\sqrt{20^{2}+80^{2}}+\sqrt{40^{2}+100^{2}}$$

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

$$3700\sqrt{10}+3760\sqrt{5}+\sqrt{4000}+\sqrt{20^{2}+80^{2}}+\sqrt{40^{2}+100^{2}}$$

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.

$$3700\sqrt{10}+3760\sqrt{5}+20\sqrt{10}+\sqrt{20^{2}+80^{2}}+\sqrt{40^{2}+100^{2}}$$

3720\sqrt{10} ni olish uchun 3700\sqrt{10} va 20\sqrt{10} ni birlashtirish.

$$3720\sqrt{10}+3760\sqrt{5}+\sqrt{20^{2}+80^{2}}+\sqrt{40^{2}+100^{2}}$$

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

$$3720\sqrt{10}+3760\sqrt{5}+\sqrt{400+80^{2}}+\sqrt{40^{2}+100^{2}}$$

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

$$3720\sqrt{10}+3760\sqrt{5}+\sqrt{400+6400}+\sqrt{40^{2}+100^{2}}$$

6800 olish uchun 400 va 6400'ni qo'shing.

$$3720\sqrt{10}+3760\sqrt{5}+\sqrt{6800}+\sqrt{40^{2}+100^{2}}$$

Faktor: 6800=20^{2}\times 17. \sqrt{20^{2}\times 17} koʻpaytmasining kvadrat ildizini \sqrt{20^{2}}\sqrt{17} kvadrat ildizlarining koʻpaytmasi sifatida qayta yozing. 20^{2} ning kvadrat ildizini chiqarish.

$$3720\sqrt{10}+3760\sqrt{5}+20\sqrt{17}+\sqrt{40^{2}+100^{2}}$$

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

$$3720\sqrt{10}+3760\sqrt{5}+20\sqrt{17}+\sqrt{1600+100^{2}}$$

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

$$3720\sqrt{10}+3760\sqrt{5}+20\sqrt{17}+\sqrt{1600+10000}$$

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

$$3720\sqrt{10}+3760\sqrt{5}+20\sqrt{17}+\sqrt{11600}$$

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.

$$3720\sqrt{10}+3760\sqrt{5}+20\sqrt{17}+20\sqrt{29}$$

Omil

$$20 {(\sqrt{17} + \sqrt{29} + 186 \sqrt{10} + 188 \sqrt{5})} = 20361,453899881$$