Algebra Examples

$\sqrt{x - 3} = \sqrt{x - 3}$

Step 1

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

${\sqrt{x - 3}}^{2} = {\sqrt{x - 3}}^{2}$

Step 2

Simplify each side of the equation.

Tap for more steps...

Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 3}$ as ${(x - 3)}^{\frac{1}{2}}$ .

${({(x - 3)}^{\frac{1}{2}})}^{2} = {\sqrt{x - 3}}^{2}$

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Simplify ${({(x - 3)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 2.2.1.1

Multiply the exponents in ${({(x - 3)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 2.2.1.1.1

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

${(x - 3)}^{\frac{1}{2} \cdot 2} = {\sqrt{x - 3}}^{2}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.2.1.1.2.1

Cancel the common factor.

${(x - 3)}^{\frac{1}{2} \cdot 2} = {\sqrt{x - 3}}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(x - 3)}^{1} = {\sqrt{x - 3}}^{2}$

Step 2.2.1.2

Simplify.

$x - 3 = {\sqrt{x - 3}}^{2}$

Step 2.3

Simplify the right side.

Tap for more steps...

Step 2.3.1

Rewrite ${\sqrt{x - 3}}^{2}$ as $x - 3$ .

Tap for more steps...

Step 2.3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 3}$ as ${(x - 3)}^{\frac{1}{2}}$ .

$x - 3 = {({(x - 3)}^{\frac{1}{2}})}^{2}$

Step 2.3.1.2

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

$x - 3 = {(x - 3)}^{\frac{1}{2} \cdot 2}$

Step 2.3.1.3

Combine $\frac{1}{2}$ and $2$ .

$x - 3 = {(x - 3)}^{\frac{2}{2}}$

Step 2.3.1.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.3.1.4.1

Cancel the common factor.

$x - 3 = {(x - 3)}^{\frac{2}{2}}$

Step 2.3.1.4.2

Rewrite the expression.

$x - 3 = {(x - 3)}^{1}$

Step 2.3.1.5

Simplify.

$x - 3 = x - 3$

Step 3

Solve for $x$ .

Tap for more steps...

Step 3.1

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

Tap for more steps...

Step 3.1.1

Subtract $x$ from both sides of the equation.

$x - 3 - x = - 3$

Step 3.1.2

Combine the opposite terms in $x - 3 - x$ .

Tap for more steps...

Step 3.1.2.1

Subtract $x$ from $x$ .

$0 - 3 = - 3$

Step 3.1.2.2

Subtract $3$ from $0$ .

$- 3 = - 3$

Step 3.2

Since $- 3 = - 3$ , the equation will always be true for any value of $x$ .

All real numbers

Step 4

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$