Solve asqrt{1/a}=sqrt{a^21/2}

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

Solve for a (complex solution)

$$a=2$$

Solution Steps

Square both sides of the equation.

$$\left(a\sqrt{\frac{1}{a}}\right)^{2}=\left(\sqrt{a^{2}\times \frac{1}{2}}\right)^{2}$$

Expand \left(a\sqrt{\frac{1}{a}}\right)^{2}.

$$a^{2}\left(\sqrt{\frac{1}{a}}\right)^{2}=\left(\sqrt{a^{2}\times \frac{1}{2}}\right)^{2}$$

Calculate \sqrt{\frac{1}{a}} to the power of 2 and get \frac{1}{a}.

$$a^{2}\times \frac{1}{a}=\left(\sqrt{a^{2}\times \frac{1}{2}}\right)^{2}$$

Express a^{2}\times \frac{1}{a} as a single fraction.

$$\frac{a^{2}}{a}=\left(\sqrt{a^{2}\times \frac{1}{2}}\right)^{2}$$

Cancel out a in both numerator and denominator.

$$a=\left(\sqrt{a^{2}\times \frac{1}{2}}\right)^{2}$$

Calculate \sqrt{a^{2}\times \frac{1}{2}} to the power of 2 and get a^{2}\times \frac{1}{2}.

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

Subtract a^{2}\times \frac{1}{2} from both sides.

$$a-a^{2}\times \frac{1}{2}=0$$

Multiply -1 and \frac{1}{2} to get -\frac{1}{2}.

$$a-\frac{1}{2}a^{2}=0$$

Factor out a.

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

To find equation solutions, solve a=0 and 1-\frac{a}{2}=0.

$$a=0 a=2$$

Substitute 0 for a in the equation a\sqrt{\frac{1}{a}}=\sqrt{a^{2}\times \frac{1}{2}}. The expression is undefined.

$$0\sqrt{\text{Indeterminate}}=\sqrt{0^{2}\times \frac{1}{2}}$$

Substitute 2 for a in the equation a\sqrt{\frac{1}{a}}=\sqrt{a^{2}\times \frac{1}{2}}.

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

Simplify. The value a=2 satisfies the equation.

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

Equation \sqrt{\frac{1}{a}}a=\sqrt{\frac{a^{2}}{2}} has a unique solution.

$$a=2$$

Solve for a

$$a=2$$

$$a=0$$

Solution Steps

Square both sides of the equation.

$$\left(a\sqrt{\frac{1}{a}}\right)^{2}=\left(\sqrt{a^{2}\times \frac{1}{2}}\right)^{2}$$

Expand \left(a\sqrt{\frac{1}{a}}\right)^{2}.

$$a^{2}\left(\sqrt{\frac{1}{a}}\right)^{2}=\left(\sqrt{a^{2}\times \frac{1}{2}}\right)^{2}$$

Calculate \sqrt{\frac{1}{a}} to the power of 2 and get \frac{1}{a}.

$$a^{2}\times \frac{1}{a}=\left(\sqrt{a^{2}\times \frac{1}{2}}\right)^{2}$$

Express a^{2}\times \frac{1}{a} as a single fraction.

$$\frac{a^{2}}{a}=\left(\sqrt{a^{2}\times \frac{1}{2}}\right)^{2}$$

Cancel out a in both numerator and denominator.

$$a=\left(\sqrt{a^{2}\times \frac{1}{2}}\right)^{2}$$

Calculate \sqrt{a^{2}\times \frac{1}{2}} to the power of 2 and get a^{2}\times \frac{1}{2}.

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

Subtract a^{2}\times \frac{1}{2} from both sides.

$$a-a^{2}\times \frac{1}{2}=0$$

Multiply -1 and \frac{1}{2} to get -\frac{1}{2}.

$$a-\frac{1}{2}a^{2}=0$$

Factor out a.

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

To find equation solutions, solve a=0 and 1-\frac{a}{2}=0.

$$a=0 a=2$$

Substitute 0 for a in the equation a\sqrt{\frac{1}{a}}=\sqrt{a^{2}\times \frac{1}{2}}. The expression is undefined.

$$0\sqrt{\text{Indeterminate}}=\sqrt{0^{2}\times \frac{1}{2}}$$

Substitute 2 for a in the equation a\sqrt{\frac{1}{a}}=\sqrt{a^{2}\times \frac{1}{2}}.

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

Simplify. The value a=2 satisfies the equation.

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

Equation \sqrt{\frac{1}{a}}a=\sqrt{\frac{a^{2}}{2}} has a unique solution.

$$a=2$$