Algebra Examples

$\sqrt{\frac{x + 3}{x + 3}} = 1$

Step 1

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

${\sqrt{\frac{x + 3}{x + 3}}}^{2} = 1^{2}$

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 $\sqrt{\frac{x + 3}{x + 3}}$ as ${(\frac{x + 3}{x + 3})}^{\frac{1}{2}}$ .

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

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

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

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Step 2.2.1.1.1

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

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

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.

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

Step 2.2.1.1.2.2

Rewrite the expression.

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

Step 2.2.1.2

Cancel the common factor of $x + 3$ .

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Step 2.2.1.2.1

Cancel the common factor.

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

Step 2.2.1.2.2

Rewrite the expression.

$1^{1} = 1^{2}$

Step 2.2.1.3

Evaluate the exponent.

$1 = 1^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

One to any power is one.

$1 = 1$

Step 3

Since $1 = 1$ , 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:

$(- \infty, \infty)$