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

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

Solve for a

$$\left\{\begin{matrix}a=\frac{17}{243}\approx 0.069958848\text{, }&b\geq 0\\a>0\text{, }&b=0\end{matrix}\right.$$

Solution Steps

Square both sides of the equation.

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

To raise \frac{\sqrt{34ab}}{\sqrt{6a}} to a power, raise both numerator and denominator to the power and then divide.

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

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

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

Calculate 9 to the power of 2 and get 81.

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

Calculate \sqrt{ab} to the power of 2 and get ab.

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

Calculate \sqrt{34ab} to the power of 2 and get 34ab.

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

Calculate \sqrt{6a} to the power of 2 and get 6a.

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

Cancel out 2a in both numerator and denominator.

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

Multiply both sides of the equation by 3.

$$17b=243ab$$

Swap sides so that all variable terms are on the left hand side.

$$243ab=17b$$

The equation is in standard form.

$$243ba=17b$$

Divide both sides by 243b.

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

Dividing by 243b undoes the multiplication by 243b.

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

Divide 17b by 243b.

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

Substitute \frac{17}{243} for a in the equation \frac{\sqrt{34ab}}{\sqrt{6a}}=9\sqrt{ab}.

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

Simplify. The value a=\frac{17}{243} satisfies the equation.

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

Equation \frac{\sqrt{34ab}}{\sqrt{6a}}=9\sqrt{ab} has a unique solution.

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

Solve for b

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

Solution Steps

Square both sides of the equation.

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

To raise \frac{\sqrt{34ab}}{\sqrt{6a}} to a power, raise both numerator and denominator to the power and then divide.

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

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

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

Calculate 9 to the power of 2 and get 81.

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

Calculate \sqrt{ab} to the power of 2 and get ab.

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

Calculate \sqrt{34ab} to the power of 2 and get 34ab.

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

Calculate \sqrt{6a} to the power of 2 and get 6a.

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

Cancel out 2a in both numerator and denominator.

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

Multiply both sides of the equation by 3.

$$17b=243ab$$

Subtract 243ab from both sides.

$$17b-243ab=0$$

Combine all terms containing b.

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

Divide 0 by 17-243a.

$$b=0$$

Substitute 0 for b in the equation \frac{\sqrt{34ab}}{\sqrt{6a}}=9\sqrt{ab}.

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

Simplify. The value b=0 satisfies the equation.

$$0=0$$

Equation \frac{\sqrt{34ab}}{\sqrt{6a}}=9\sqrt{ab} has a unique solution.

$$b=0$$