Solve sqrt{ab}=sqrt{a/b}

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

Solve for a (complex solution)

$$\left\{\begin{matrix}a\neq 0\text{, }&b=-1\text{ or }b=1\\a=0\text{, }&b\neq 0\end{matrix}\right.$$

Solution Steps

Square both sides of the equation.

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

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

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

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

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

Multiply both sides of the equation by b.

$$abb=a$$

Multiply b and b to get b^{2}.

$$ab^{2}=a$$

Subtract a from both sides.

$$ab^{2}-a=0$$

Combine all terms containing a.

$$\left(b^{2}-1\right)a=0$$

Divide 0 by b^{2}-1.

$$a=0$$

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

$$\sqrt{0b}=\sqrt{\frac{0}{b}}$$

Simplify. The value a=0 satisfies the equation.

$$0=0$$

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

$$a=0$$

Solve for a

$$\left\{\begin{matrix}a=0\text{, }&b\neq 0\text{ and }|b|\neq 1\\a\leq 0\text{, }&b=-1\\a\geq 0\text{, }&b=1\end{matrix}\right.$$

Solution Steps

Square both sides of the equation.

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

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

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

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

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

Multiply both sides of the equation by b.

$$abb=a$$

Multiply b and b to get b^{2}.

$$ab^{2}=a$$

Subtract a from both sides.

$$ab^{2}-a=0$$

Combine all terms containing a.

$$\left(b^{2}-1\right)a=0$$

Divide 0 by b^{2}-1.

$$a=0$$

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

$$\sqrt{0b}=\sqrt{\frac{0}{b}}$$

Simplify. The value a=0 satisfies the equation.

$$0=0$$

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

$$a=0$$

Solve for b (complex solution)

$$\left\{\begin{matrix}\\b=-1\text{; }b=1\text{, }&\text{unconditionally}\\b\neq 0\text{, }&a=0\end{matrix}\right.$$

Solve for b

$$\left\{\begin{matrix}b=-1\text{, }&a<0\\b=1\text{, }&a>0\\b\neq 0\text{, }&a=0\end{matrix}\right.$$