frac{21sqrt{15}}{512+5sqrt{3}} eching

$$\frac { 21 \sqrt { 15 } } { 512 + 5 \sqrt { 3 } }$$

Baholash

$$\frac{10752\sqrt{15}-315\sqrt{5}}{262069}\approx 0,156210599$$

Yechim qadamlari

\frac{21\sqrt{15}}{512+5\sqrt{3}} maxrajini 512-5\sqrt{3} orqali surat va maxrajini koʻpaytirish orqali ratsionallashtiring.

$$\frac{21\sqrt{15}\left(512-5\sqrt{3}\right)}{\left(512+5\sqrt{3}\right)\left(512-5\sqrt{3}\right)}$$

Hisoblang: \left(512+5\sqrt{3}\right)\left(512-5\sqrt{3}\right). Ko‘paytirish qoida yordamida turli kvadratlarga aylantirilishi mumkin: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{21\sqrt{15}\left(512-5\sqrt{3}\right)}{512^{2}-\left(5\sqrt{3}\right)^{2}}$$

2 daraja ko‘rsatkichini 512 ga hisoblang va 262144 ni qiymatni oling.

$$\frac{21\sqrt{15}\left(512-5\sqrt{3}\right)}{262144-\left(5\sqrt{3}\right)^{2}}$$

\left(5\sqrt{3}\right)^{2} ni kengaytirish.

$$\frac{21\sqrt{15}\left(512-5\sqrt{3}\right)}{262144-5^{2}\left(\sqrt{3}\right)^{2}}$$

2 daraja ko‘rsatkichini 5 ga hisoblang va 25 ni qiymatni oling.

$$\frac{21\sqrt{15}\left(512-5\sqrt{3}\right)}{262144-25\left(\sqrt{3}\right)^{2}}$$

\sqrt{3} kvadrati – 3.

$$\frac{21\sqrt{15}\left(512-5\sqrt{3}\right)}{262144-25\times 3}$$

75 hosil qilish uchun 25 va 3 ni ko'paytirish.

$$\frac{21\sqrt{15}\left(512-5\sqrt{3}\right)}{262144-75}$$

262069 olish uchun 262144 dan 75 ni ayirish.

$$\frac{21\sqrt{15}\left(512-5\sqrt{3}\right)}{262069}$$

21\sqrt{15} ga 512-5\sqrt{3} ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\frac{10752\sqrt{15}-105\sqrt{3}\sqrt{15}}{262069}$$

Faktor: 15=3\times 5. \sqrt{3\times 5} koʻpaytmasining kvadrat ildizini \sqrt{3}\sqrt{5} kvadrat ildizlarining koʻpaytmasi sifatida qayta yozing.

$$\frac{10752\sqrt{15}-105\sqrt{3}\sqrt{3}\sqrt{5}}{262069}$$

3 hosil qilish uchun \sqrt{3} va \sqrt{3} ni ko'paytirish.

$$\frac{10752\sqrt{15}-105\times 3\sqrt{5}}{262069}$$

-315 hosil qilish uchun -105 va 3 ni ko'paytirish.

$$\frac{10752\sqrt{15}-315\sqrt{5}}{262069}$$