frac{sqrt{34ab}}{sqrt{6a}}=9sqrt{ab} eching

$$\frac { \sqrt { 34 a b } } { \sqrt { 6 a } } = 9 \sqrt { a b }$$

a uchun yechish (complex solution)

$$\left\{\begin{matrix}\\a=\frac{17}{243}\approx 0,069958848\text{, }&\text{unconditionally}\\a\neq 0\text{, }&b=0\end{matrix}\right,$$

Yechim qadamlari

Tenglamaning ikkala taraf kvadratini chiqarish.

$$\left(\frac{\sqrt{34ab}}{\sqrt{6a}}\right)^{2}=\left(9\sqrt{ab}\right)^{2}$$

\frac{\sqrt{34ab}}{\sqrt{6a}}ni darajaga oshirish uchun, surat va maxrajni darajaga oshirib, keyin bo‘ling.

$$\frac{\left(\sqrt{34ab}\right)^{2}}{\left(\sqrt{6a}\right)^{2}}=\left(9\sqrt{ab}\right)^{2}$$

\left(9\sqrt{ab}\right)^{2} ni kengaytirish.

$$\frac{\left(\sqrt{34ab}\right)^{2}}{\left(\sqrt{6a}\right)^{2}}=9^{2}\left(\sqrt{ab}\right)^{2}$$

2 daraja ko‘rsatkichini 9 ga hisoblang va 81 ni qiymatni oling.

$$\frac{\left(\sqrt{34ab}\right)^{2}}{\left(\sqrt{6a}\right)^{2}}=81\left(\sqrt{ab}\right)^{2}$$

2 daraja ko‘rsatkichini \sqrt{ab} ga hisoblang va ab ni qiymatni oling.

$$\frac{\left(\sqrt{34ab}\right)^{2}}{\left(\sqrt{6a}\right)^{2}}=81ab$$

2 daraja ko‘rsatkichini \sqrt{34ab} ga hisoblang va 34ab ni qiymatni oling.

$$\frac{34ab}{\left(\sqrt{6a}\right)^{2}}=81ab$$

2 daraja ko‘rsatkichini \sqrt{6a} ga hisoblang va 6a ni qiymatni oling.

$$\frac{34ab}{6a}=81ab$$

Surat va maxrajdagi ikkala 2a ni qisqartiring.

$$\frac{17b}{3}=81ab$$

Tenglamaning ikkala tarafini 3 ga ko'paytirish.

$$17b=243ab$$

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$243ab=17b$$

Tenglama standart shaklda.

$$243ba=17b$$

Ikki tarafini 243b ga bo‘ling.

$$\frac{243ba}{243b}=\frac{17b}{243b}$$

243b ga bo'lish 243b ga ko'paytirishni bekor qiladi.

$$a=\frac{17b}{243b}$$

17b ni 243b ga bo'lish.

$$a=\frac{17}{243}$$

\frac{\sqrt{34ab}}{\sqrt{6a}}=9\sqrt{ab} tenglamasida a uchun \frac{17}{243} ni almashtiring.

$$\frac{\sqrt{34\times \frac{17}{243}b}}{\sqrt{6\times \frac{17}{243}}}=9\sqrt{\frac{17}{243}b}$$

Qisqartirish. a=\frac{17}{243} tenglamani qoniqtiradi.

$$\frac{1}{3}b^{\frac{1}{2}}\times 51^{\frac{1}{2}}=\frac{1}{3}b^{\frac{1}{2}}\times 51^{\frac{1}{2}}$$

\frac{\sqrt{34ab}}{\sqrt{6a}}=9\sqrt{ab} tenglamasi noyob yechimga ega.

$$a=\frac{17}{243}$$

b uchun yechish (complex solution)

$$\left\{\begin{matrix}b=0\text{, }&a\neq 0\\b\neq 0\text{, }&a=\frac{17}{243}\end{matrix}\right,$$

Yechim qadamlari

Tenglamaning ikkala taraf kvadratini chiqarish.

$$\left(\frac{\sqrt{34ab}}{\sqrt{6a}}\right)^{2}=\left(9\sqrt{ab}\right)^{2}$$

\frac{\sqrt{34ab}}{\sqrt{6a}}ni darajaga oshirish uchun, surat va maxrajni darajaga oshirib, keyin bo‘ling.

$$\frac{\left(\sqrt{34ab}\right)^{2}}{\left(\sqrt{6a}\right)^{2}}=\left(9\sqrt{ab}\right)^{2}$$

\left(9\sqrt{ab}\right)^{2} ni kengaytirish.

$$\frac{\left(\sqrt{34ab}\right)^{2}}{\left(\sqrt{6a}\right)^{2}}=9^{2}\left(\sqrt{ab}\right)^{2}$$

2 daraja ko‘rsatkichini 9 ga hisoblang va 81 ni qiymatni oling.

$$\frac{\left(\sqrt{34ab}\right)^{2}}{\left(\sqrt{6a}\right)^{2}}=81\left(\sqrt{ab}\right)^{2}$$

2 daraja ko‘rsatkichini \sqrt{ab} ga hisoblang va ab ni qiymatni oling.

$$\frac{\left(\sqrt{34ab}\right)^{2}}{\left(\sqrt{6a}\right)^{2}}=81ab$$

2 daraja ko‘rsatkichini \sqrt{34ab} ga hisoblang va 34ab ni qiymatni oling.

$$\frac{34ab}{\left(\sqrt{6a}\right)^{2}}=81ab$$

2 daraja ko‘rsatkichini \sqrt{6a} ga hisoblang va 6a ni qiymatni oling.

$$\frac{34ab}{6a}=81ab$$

Surat va maxrajdagi ikkala 2a ni qisqartiring.

$$\frac{17b}{3}=81ab$$

Tenglamaning ikkala tarafini 3 ga ko'paytirish.

$$17b=243ab$$

Ikkala tarafdan 243ab ni ayirish.

$$17b-243ab=0$$

b'ga ega bo'lgan barcha shartlarni birlashtirish.

$$\left(17-243a\right)b=0$$

0 ni 17-243a ga bo'lish.

$$b=0$$

\frac{\sqrt{34ab}}{\sqrt{6a}}=9\sqrt{ab} tenglamasida b uchun 0 ni almashtiring.

$$\frac{\sqrt{34a\times 0}}{\sqrt{6a}}=9\sqrt{a\times 0}$$

Qisqartirish. b=0 tenglamani qoniqtiradi.

$$0=0$$

\frac{\sqrt{34ab}}{\sqrt{6a}}=9\sqrt{ab} tenglamasi noyob yechimga ega.

$$b=0$$

a uchun yechish

$$\left\{\begin{matrix}a=\frac{17}{243}\approx 0,069958848\text{, }&b\geq 0\\a>0\text{, }&b=0\end{matrix}\right,$$

Yechim qadamlari

Tenglamaning ikkala taraf kvadratini chiqarish.

$$\left(\frac{\sqrt{34ab}}{\sqrt{6a}}\right)^{2}=\left(9\sqrt{ab}\right)^{2}$$

\frac{\sqrt{34ab}}{\sqrt{6a}}ni darajaga oshirish uchun, surat va maxrajni darajaga oshirib, keyin bo‘ling.

$$\frac{\left(\sqrt{34ab}\right)^{2}}{\left(\sqrt{6a}\right)^{2}}=\left(9\sqrt{ab}\right)^{2}$$

\left(9\sqrt{ab}\right)^{2} ni kengaytirish.

$$\frac{\left(\sqrt{34ab}\right)^{2}}{\left(\sqrt{6a}\right)^{2}}=9^{2}\left(\sqrt{ab}\right)^{2}$$

2 daraja ko‘rsatkichini 9 ga hisoblang va 81 ni qiymatni oling.

$$\frac{\left(\sqrt{34ab}\right)^{2}}{\left(\sqrt{6a}\right)^{2}}=81\left(\sqrt{ab}\right)^{2}$$

2 daraja ko‘rsatkichini \sqrt{ab} ga hisoblang va ab ni qiymatni oling.

$$\frac{\left(\sqrt{34ab}\right)^{2}}{\left(\sqrt{6a}\right)^{2}}=81ab$$

2 daraja ko‘rsatkichini \sqrt{34ab} ga hisoblang va 34ab ni qiymatni oling.

$$\frac{34ab}{\left(\sqrt{6a}\right)^{2}}=81ab$$

2 daraja ko‘rsatkichini \sqrt{6a} ga hisoblang va 6a ni qiymatni oling.

$$\frac{34ab}{6a}=81ab$$

Surat va maxrajdagi ikkala 2a ni qisqartiring.

$$\frac{17b}{3}=81ab$$

Tenglamaning ikkala tarafini 3 ga ko'paytirish.

$$17b=243ab$$

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$243ab=17b$$

Tenglama standart shaklda.

$$243ba=17b$$

Ikki tarafini 243b ga bo‘ling.

$$\frac{243ba}{243b}=\frac{17b}{243b}$$

243b ga bo'lish 243b ga ko'paytirishni bekor qiladi.

$$a=\frac{17b}{243b}$$

17b ni 243b ga bo'lish.

$$a=\frac{17}{243}$$

\frac{\sqrt{34ab}}{\sqrt{6a}}=9\sqrt{ab} tenglamasida a uchun \frac{17}{243} ni almashtiring.

$$\frac{\sqrt{34\times \frac{17}{243}b}}{\sqrt{6\times \frac{17}{243}}}=9\sqrt{\frac{17}{243}b}$$

Qisqartirish. a=\frac{17}{243} tenglamani qoniqtiradi.

$$\frac{1}{3}b^{\frac{1}{2}}\times 51^{\frac{1}{2}}=\frac{1}{3}b^{\frac{1}{2}}\times 51^{\frac{1}{2}}$$

\frac{\sqrt{34ab}}{\sqrt{6a}}=9\sqrt{ab} tenglamasi noyob yechimga ega.

$$a=\frac{17}{243}$$

b uchun yechish

$$\left\{\begin{matrix}b=0\text{, }&a>0\\b\geq 0\text{, }&a=\frac{17}{243}\end{matrix}\right,$$

Yechim qadamlari

Tenglamaning ikkala taraf kvadratini chiqarish.

$$\left(\frac{\sqrt{34ab}}{\sqrt{6a}}\right)^{2}=\left(9\sqrt{ab}\right)^{2}$$

\frac{\sqrt{34ab}}{\sqrt{6a}}ni darajaga oshirish uchun, surat va maxrajni darajaga oshirib, keyin bo‘ling.

$$\frac{\left(\sqrt{34ab}\right)^{2}}{\left(\sqrt{6a}\right)^{2}}=\left(9\sqrt{ab}\right)^{2}$$

\left(9\sqrt{ab}\right)^{2} ni kengaytirish.

$$\frac{\left(\sqrt{34ab}\right)^{2}}{\left(\sqrt{6a}\right)^{2}}=9^{2}\left(\sqrt{ab}\right)^{2}$$

2 daraja ko‘rsatkichini 9 ga hisoblang va 81 ni qiymatni oling.

$$\frac{\left(\sqrt{34ab}\right)^{2}}{\left(\sqrt{6a}\right)^{2}}=81\left(\sqrt{ab}\right)^{2}$$

2 daraja ko‘rsatkichini \sqrt{ab} ga hisoblang va ab ni qiymatni oling.

$$\frac{\left(\sqrt{34ab}\right)^{2}}{\left(\sqrt{6a}\right)^{2}}=81ab$$

2 daraja ko‘rsatkichini \sqrt{34ab} ga hisoblang va 34ab ni qiymatni oling.

$$\frac{34ab}{\left(\sqrt{6a}\right)^{2}}=81ab$$

2 daraja ko‘rsatkichini \sqrt{6a} ga hisoblang va 6a ni qiymatni oling.

$$\frac{34ab}{6a}=81ab$$

Surat va maxrajdagi ikkala 2a ni qisqartiring.

$$\frac{17b}{3}=81ab$$

Tenglamaning ikkala tarafini 3 ga ko'paytirish.

$$17b=243ab$$

Ikkala tarafdan 243ab ni ayirish.

$$17b-243ab=0$$

b'ga ega bo'lgan barcha shartlarni birlashtirish.

$$\left(17-243a\right)b=0$$

0 ni 17-243a ga bo'lish.

$$b=0$$

\frac{\sqrt{34ab}}{\sqrt{6a}}=9\sqrt{ab} tenglamasida b uchun 0 ni almashtiring.

$$\frac{\sqrt{34a\times 0}}{\sqrt{6a}}=9\sqrt{a\times 0}$$

Qisqartirish. b=0 tenglamani qoniqtiradi.

$$0=0$$

\frac{\sqrt{34ab}}{\sqrt{6a}}=9\sqrt{ab} tenglamasi noyob yechimga ega.

$$b=0$$