Solve sqrt{a^3b}=sqrt{b/a}

$$\sqrt { a ^ { 3 } b } = \sqrt { \frac { b } { a } }$$

Solve for b (complex solution)

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

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{a^{3}b}\right)^{2}=\left(\sqrt{\frac{b}{a}}\right)^{2}$$

Calculate \sqrt{a^{3}b} to the power of 2 and get a^{3}b.

$$a^{3}b=\left(\sqrt{\frac{b}{a}}\right)^{2}$$

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

$$a^{3}b=\frac{b}{a}$$

Multiply both sides of the equation by a.

$$a^{3}ba=b$$

To multiply powers of the same base, add their exponents. Add 3 and 1 to get 4.

$$a^{4}b=b$$

Subtract b from both sides.

$$a^{4}b-b=0$$

Combine all terms containing b.

$$\left(a^{4}-1\right)b=0$$

Divide 0 by a^{4}-1.

$$b=0$$

Substitute 0 for b in the equation \sqrt{a^{3}b}=\sqrt{\frac{b}{a}}.

$$\sqrt{a^{3}\times 0}=\sqrt{\frac{0}{a}}$$

Simplify. The value b=0 satisfies the equation.

$$0=0$$

Equation \sqrt{ba^{3}}=\sqrt{\frac{b}{a}} has a unique solution.

$$b=0$$

Solve for b

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

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{a^{3}b}\right)^{2}=\left(\sqrt{\frac{b}{a}}\right)^{2}$$

Calculate \sqrt{a^{3}b} to the power of 2 and get a^{3}b.

$$a^{3}b=\left(\sqrt{\frac{b}{a}}\right)^{2}$$

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

$$a^{3}b=\frac{b}{a}$$

Multiply both sides of the equation by a.

$$a^{3}ba=b$$

To multiply powers of the same base, add their exponents. Add 3 and 1 to get 4.

$$a^{4}b=b$$

Subtract b from both sides.

$$a^{4}b-b=0$$

Combine all terms containing b.

$$\left(a^{4}-1\right)b=0$$

Divide 0 by a^{4}-1.

$$b=0$$

Substitute 0 for b in the equation \sqrt{a^{3}b}=\sqrt{\frac{b}{a}}.

$$\sqrt{a^{3}\times 0}=\sqrt{\frac{0}{a}}$$

Simplify. The value b=0 satisfies the equation.

$$0=0$$

Equation \sqrt{ba^{3}}=\sqrt{\frac{b}{a}} has a unique solution.

$$b=0$$

Solve for a (complex solution)

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

Solve for a

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