Algebra Examples

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

Step 1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{x}}^{2} = {(x^{\frac{1}{2}})}^{2}$

Step 2

Simplify each side of the equation.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

${(x^{\frac{1}{2}})}^{2} = {(x^{\frac{1}{2}})}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${(x^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{\frac{1}{2} \cdot 2} = {(x^{\frac{1}{2}})}^{2}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = {(x^{\frac{1}{2}})}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

$x^{1} = {(x^{\frac{1}{2}})}^{2}$

Step 2.2.1.2

Simplify.

$x = {(x^{\frac{1}{2}})}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${(x^{\frac{1}{2}})}^{2}$ .

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Step 2.3.1.1

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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Step 2.3.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.3.1.1.2

Cancel the common factor of $2$ .

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Step 2.3.1.1.2.1

Cancel the common factor.

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

Step 2.3.1.1.2.2

Rewrite the expression.

$x = x^{1}$

Step 2.3.1.2

Simplify.

$x = x$

Step 3

Solve for $x$ .

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Step 3.1

Move all terms containing $x$ to the left side of the equation.

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Step 3.1.1

Subtract $x$ from both sides of the equation.

$x - x = 0$

Step 3.1.2

Subtract $x$ from $x$ .

$0 = 0$

Step 3.2

Since $0 = 0$ , the equation will always be true.

Always true

Step 4

The result can be shown in multiple forms.

Always true

Interval Notation:

$(- \infty, \infty)$