Solve sqrt{a*b}=sqrt{a}

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

Solve for a (complex solution)

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

Solution Steps

Square both sides of the equation.

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

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

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

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

$$ab=a$$

Subtract a from both sides.

$$ab-a=0$$

Combine all terms containing a.

$$\left(b-1\right)a=0$$

Divide 0 by -1+b.

$$a=0$$

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

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

Simplify. The value a=0 satisfies the equation.

$$0=0$$

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

$$a=0$$

Solve for a

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

Solution Steps

Square both sides of the equation.

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

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

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

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

$$ab=a$$

Subtract a from both sides.

$$ab-a=0$$

Combine all terms containing a.

$$\left(b-1\right)a=0$$

Divide 0 by b-1.

$$a=0$$

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

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

Simplify. The value a=0 satisfies the equation.

$$0=0$$

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

$$a=0$$

Solve for b

$$\left\{\begin{matrix}b=1\text{, }&a\geq 0\\b\in \mathrm{R}\text{, }&a=0\end{matrix}\right.$$

Steps for Solving Radical Equations

Square both sides of the equation.

$$ab=a$$

Divide both sides by a.

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

Dividing by a undoes the multiplication by a.

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

Divide a by a.

$$b=1$$

Solve for b (complex solution)

$$\left\{\begin{matrix}\\b=1\text{, }&\text{unconditionally}\\b\in \mathrm{C}\text{, }&a=0\end{matrix}\right.$$