Algebra Examples

$\sqrt{v - 5} = \sqrt{23 - v}$

Step 1

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

${\sqrt{v - 5}}^{2} = {\sqrt{23 - v}}^{2}$

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 $\sqrt{v - 5}$ as ${(v - 5)}^{\frac{1}{2}}$ .

${({(v - 5)}^{\frac{1}{2}})}^{2} = {\sqrt{23 - v}}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(v - 5)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(v - 5)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(v - 5)}^{\frac{1}{2} \cdot 2} = {\sqrt{23 - v}}^{2}$

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.

${(v - 5)}^{\frac{1}{2} \cdot 2} = {\sqrt{23 - v}}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(v - 5)}^{1} = {\sqrt{23 - v}}^{2}$

Step 2.2.1.2

Simplify.

$v - 5 = {\sqrt{23 - v}}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Rewrite ${\sqrt{23 - v}}^{2}$ as $23 - v$ .

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Step 2.3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{23 - v}$ as ${(23 - v)}^{\frac{1}{2}}$ .

$v - 5 = {({(23 - v)}^{\frac{1}{2}})}^{2}$

Step 2.3.1.2

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

$v - 5 = {(23 - v)}^{\frac{1}{2} \cdot 2}$

Step 2.3.1.3

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

$v - 5 = {(23 - v)}^{\frac{2}{2}}$

Step 2.3.1.4

Cancel the common factor of $2$ .

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Step 2.3.1.4.1

Cancel the common factor.

$v - 5 = {(23 - v)}^{\frac{2}{2}}$

Step 2.3.1.4.2

Rewrite the expression.

$v - 5 = {(23 - v)}^{1}$

Step 2.3.1.5

Simplify.

$v - 5 = 23 - v$

Step 3

Solve for $v$ .

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Step 3.1

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

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Step 3.1.1

Add $v$ to both sides of the equation.

$v - 5 + v = 23$

Step 3.1.2

Add $v$ and $v$ .

$2 v - 5 = 23$

Step 3.2

Move all terms not containing $v$ to the right side of the equation.

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Step 3.2.1

Add $5$ to both sides of the equation.

$2 v = 23 + 5$

Step 3.2.2

Add $23$ and $5$ .

$2 v = 28$

Step 3.3

Divide each term in $2 v = 28$ by $2$ and simplify.

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Step 3.3.1

Divide each term in $2 v = 28$ by $2$ .

$\frac{2 v}{2} = \frac{28}{2}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $2$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{2 v}{2} = \frac{28}{2}$

Step 3.3.2.1.2

Divide $v$ by $1$ .

$v = \frac{28}{2}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Divide $28$ by $2$ .

$v = 14$