Solve sqrt{2a/b^2}=sqrt{a/b}

$$\sqrt { \frac { 2 a } { b ^ { 2 } } } = \sqrt { \frac { a } { b } }$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a\neq 0\text{, }&b=2\\a=0\text{, }&b\neq 0\end{matrix}\right.$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{\frac{2a}{b^{2}}}\right)^{2}=\left(\sqrt{\frac{a}{b}}\right)^{2}$$

Calculate \sqrt{\frac{2a}{b^{2}}} to the power of 2 and get \frac{2a}{b^{2}}.

$$\frac{2a}{b^{2}}=\left(\sqrt{\frac{a}{b}}\right)^{2}$$

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

$$\frac{2a}{b^{2}}=\frac{a}{b}$$

Multiply both sides of the equation by b^{2}, the least common multiple of b^{2},b.

$$2a=ba$$

Subtract ba from both sides.

$$2a-ba=0$$

Combine all terms containing a.

$$\left(2-b\right)a=0$$

Divide 0 by 2-b.

$$a=0$$

Substitute 0 for a in the equation \sqrt{\frac{2a}{b^{2}}}=\sqrt{\frac{a}{b}}.

$$\sqrt{\frac{2\times 0}{b^{2}}}=\sqrt{\frac{0}{b}}$$

Simplify. The value a=0 satisfies the equation.

$$0=0$$

Equation \sqrt{\frac{2a}{b^{2}}}=\sqrt{\frac{a}{b}} has a unique solution.

$$a=0$$

Solve for b (complex solution)

$$\left\{\begin{matrix}\\b=2\text{, }&\text{unconditionally}\\b\neq 0\text{, }&a=0\end{matrix}\right.$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{\frac{2a}{b^{2}}}\right)^{2}=\left(\sqrt{\frac{a}{b}}\right)^{2}$$

Calculate \sqrt{\frac{2a}{b^{2}}} to the power of 2 and get \frac{2a}{b^{2}}.

$$\frac{2a}{b^{2}}=\left(\sqrt{\frac{a}{b}}\right)^{2}$$

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

$$\frac{2a}{b^{2}}=\frac{a}{b}$$

Multiply both sides of the equation by b^{2}, the least common multiple of b^{2},b.

$$2a=ba$$

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

$$ba=2a$$

The equation is in standard form.

$$ab=2a$$

Divide both sides by a.

$$\frac{ab}{a}=\frac{2a}{a}$$

Dividing by a undoes the multiplication by a.

$$b=\frac{2a}{a}$$

Divide 2a by a.

$$b=2$$

Substitute 2 for b in the equation \sqrt{\frac{2a}{b^{2}}}=\sqrt{\frac{a}{b}}.

$$\sqrt{\frac{2a}{2^{2}}}=\sqrt{\frac{a}{2}}$$

Simplify. The value b=2 satisfies the equation.

$$\frac{1}{2}\times 2^{\frac{1}{2}}a^{\frac{1}{2}}=\frac{1}{2}\times 2^{\frac{1}{2}}a^{\frac{1}{2}}$$

Equation \sqrt{\frac{2a}{b^{2}}}=\sqrt{\frac{a}{b}} has a unique solution.

$$b=2$$

Solve for a

$$\left\{\begin{matrix}a=0\text{, }&b\neq 2\text{ and }b\neq 0\\a\geq 0\text{, }&b=2\end{matrix}\right.$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{\frac{2a}{b^{2}}}\right)^{2}=\left(\sqrt{\frac{a}{b}}\right)^{2}$$

Calculate \sqrt{\frac{2a}{b^{2}}} to the power of 2 and get \frac{2a}{b^{2}}.

$$\frac{2a}{b^{2}}=\left(\sqrt{\frac{a}{b}}\right)^{2}$$

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

$$\frac{2a}{b^{2}}=\frac{a}{b}$$

Multiply both sides of the equation by b^{2}, the least common multiple of b^{2},b.

$$2a=ba$$

Subtract ba from both sides.

$$2a-ba=0$$

Combine all terms containing a.

$$\left(2-b\right)a=0$$

Divide 0 by 2-b.

$$a=0$$

Substitute 0 for a in the equation \sqrt{\frac{2a}{b^{2}}}=\sqrt{\frac{a}{b}}.

$$\sqrt{\frac{2\times 0}{b^{2}}}=\sqrt{\frac{0}{b}}$$

Simplify. The value a=0 satisfies the equation.

$$0=0$$

Equation \sqrt{\frac{2a}{b^{2}}}=\sqrt{\frac{a}{b}} has a unique solution.

$$a=0$$

Solve for b

$$\left\{\begin{matrix}b=2\text{, }&a>0\\b\neq 0\text{, }&a=0\end{matrix}\right.$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{\frac{2a}{b^{2}}}\right)^{2}=\left(\sqrt{\frac{a}{b}}\right)^{2}$$

Calculate \sqrt{\frac{2a}{b^{2}}} to the power of 2 and get \frac{2a}{b^{2}}.

$$\frac{2a}{b^{2}}=\left(\sqrt{\frac{a}{b}}\right)^{2}$$

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

$$\frac{2a}{b^{2}}=\frac{a}{b}$$

Multiply both sides of the equation by b^{2}, the least common multiple of b^{2},b.

$$2a=ba$$

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

$$ba=2a$$

The equation is in standard form.

$$ab=2a$$

Divide both sides by a.

$$\frac{ab}{a}=\frac{2a}{a}$$

Dividing by a undoes the multiplication by a.

$$b=\frac{2a}{a}$$

Divide 2a by a.

$$b=2$$

Substitute 2 for b in the equation \sqrt{\frac{2a}{b^{2}}}=\sqrt{\frac{a}{b}}.

$$\sqrt{\frac{2a}{2^{2}}}=\sqrt{\frac{a}{2}}$$

Simplify. The value b=2 satisfies the equation.

$$\frac{1}{2}\times 2^{\frac{1}{2}}a^{\frac{1}{2}}=\frac{1}{2}\times 2^{\frac{1}{2}}a^{\frac{1}{2}}$$

Equation \sqrt{\frac{2a}{b^{2}}}=\sqrt{\frac{a}{b}} has a unique solution.

$$b=2$$