Algebra Examples

$\sqrt{2 v - 5} = \sqrt{\frac{v}{3} + 5}$

Step 1

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

${\sqrt{2 v - 5}}^{2} = {\sqrt{\frac{v}{3} + 5}}^{2}$

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

${({(2 v - 5)}^{\frac{1}{2}})}^{2} = {\sqrt{\frac{v}{3} + 5}}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

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

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Step 2.2.1.1.1

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

${(2 v - 5)}^{\frac{1}{2} \cdot 2} = {\sqrt{\frac{v}{3} + 5}}^{2}$

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.

${(2 v - 5)}^{\frac{1}{2} \cdot 2} = {\sqrt{\frac{v}{3} + 5}}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(2 v - 5)}^{1} = {\sqrt{\frac{v}{3} + 5}}^{2}$

Step 2.2.1.2

Simplify.

$2 v - 5 = {\sqrt{\frac{v}{3} + 5}}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${\sqrt{\frac{v}{3} + 5}}^{2}$ .

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Step 2.3.1.1

To write $5$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$2 v - 5 = {\sqrt{\frac{v}{3} + 5 \cdot \frac{3}{3}}}^{2}$

Step 2.3.1.2

Combine $5$ and $\frac{3}{3}$ .

$2 v - 5 = {\sqrt{\frac{v}{3} + \frac{5 \cdot 3}{3}}}^{2}$

Step 2.3.1.3

Combine the numerators over the common denominator.

$2 v - 5 = {\sqrt{\frac{v + 5 \cdot 3}{3}}}^{2}$

Step 2.3.1.4

Multiply $5$ by $3$ .

$2 v - 5 = {\sqrt{\frac{v + 15}{3}}}^{2}$

Step 2.3.1.5

Rewrite $\sqrt{\frac{v + 15}{3}}$ as $\frac{\sqrt{v + 15}}{\sqrt{3}}$ .

$2 v - 5 = {(\frac{\sqrt{v + 15}}{\sqrt{3}})}^{2}$

Step 2.3.1.6

Multiply $\frac{\sqrt{v + 15}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$2 v - 5 = {(\frac{\sqrt{v + 15}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})}^{2}$

Step 2.3.1.7

Simplify terms.

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Step 2.3.1.7.1

Combine and simplify the denominator.

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Step 2.3.1.7.1.1

Multiply $\frac{\sqrt{v + 15}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$2 v - 5 = {(\frac{\sqrt{v + 15} \sqrt{3}}{\sqrt{3} \sqrt{3}})}^{2}$

Step 2.3.1.7.1.2

Raise $\sqrt{3}$ to the power of $1$ .

$2 v - 5 = {(\frac{\sqrt{v + 15} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}})}^{2}$

Step 2.3.1.7.1.3

Raise $\sqrt{3}$ to the power of $1$ .

$2 v - 5 = {(\frac{\sqrt{v + 15} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}})}^{2}$

Step 2.3.1.7.1.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 v - 5 = {(\frac{\sqrt{v + 15} \sqrt{3}}{{\sqrt{3}}^{1 + 1}})}^{2}$

Step 2.3.1.7.1.5

Add $1$ and $1$ .

$2 v - 5 = {(\frac{\sqrt{v + 15} \sqrt{3}}{{\sqrt{3}}^{2}})}^{2}$

Step 2.3.1.7.1.6

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

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Step 2.3.1.7.1.6.1

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

$2 v - 5 = {(\frac{\sqrt{v + 15} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}})}^{2}$

Step 2.3.1.7.1.6.2

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

$2 v - 5 = {(\frac{\sqrt{v + 15} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}})}^{2}$

Step 2.3.1.7.1.6.3

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

$2 v - 5 = {(\frac{\sqrt{v + 15} \sqrt{3}}{3^{\frac{2}{2}}})}^{2}$

Step 2.3.1.7.1.6.4

Cancel the common factor of $2$ .

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Step 2.3.1.7.1.6.4.1

Cancel the common factor.

$2 v - 5 = {(\frac{\sqrt{v + 15} \sqrt{3}}{3^{\frac{2}{2}}})}^{2}$

Step 2.3.1.7.1.6.4.2

Rewrite the expression.

$2 v - 5 = {(\frac{\sqrt{v + 15} \sqrt{3}}{3^{1}})}^{2}$

Step 2.3.1.7.1.6.5

Evaluate the exponent.

$2 v - 5 = {(\frac{\sqrt{v + 15} \sqrt{3}}{3})}^{2}$

Step 2.3.1.7.2

Combine using the product rule for radicals.

$2 v - 5 = {(\frac{\sqrt{(v + 15) \cdot 3}}{3})}^{2}$

Step 2.3.1.7.3

Apply the product rule to $\frac{\sqrt{(v + 15) \cdot 3}}{3}$ .

$2 v - 5 = \frac{{\sqrt{(v + 15) \cdot 3}}^{2}}{3^{2}}$

Step 2.3.1.8

Simplify the numerator.

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Step 2.3.1.8.1

Rewrite ${\sqrt{(v + 15) \cdot 3}}^{2}$ as $(v + 15) \cdot 3$ .

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Step 2.3.1.8.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(v + 15) \cdot 3}$ as ${((v + 15) \cdot 3)}^{\frac{1}{2}}$ .

$2 v - 5 = \frac{{({((v + 15) \cdot 3)}^{\frac{1}{2}})}^{2}}{3^{2}}$

Step 2.3.1.8.1.2

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

$2 v - 5 = \frac{{((v + 15) \cdot 3)}^{\frac{1}{2} \cdot 2}}{3^{2}}$

Step 2.3.1.8.1.3

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

$2 v - 5 = \frac{{((v + 15) \cdot 3)}^{\frac{2}{2}}}{3^{2}}$

Step 2.3.1.8.1.4

Cancel the common factor of $2$ .

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Step 2.3.1.8.1.4.1

Cancel the common factor.

$2 v - 5 = \frac{{((v + 15) \cdot 3)}^{\frac{2}{2}}}{3^{2}}$

Step 2.3.1.8.1.4.2

Rewrite the expression.

$2 v - 5 = \frac{{((v + 15) \cdot 3)}^{1}}{3^{2}}$

Step 2.3.1.8.1.5

Simplify.

$2 v - 5 = \frac{(v + 15) \cdot 3}{3^{2}}$

Step 2.3.1.8.2

Apply the distributive property.

$2 v - 5 = \frac{v \cdot 3 + 15 \cdot 3}{3^{2}}$

Step 2.3.1.8.3

Move $3$ to the left of $v$ .

$2 v - 5 = \frac{3 \cdot v + 15 \cdot 3}{3^{2}}$

Step 2.3.1.8.4

Multiply $15$ by $3$ .

$2 v - 5 = \frac{3 \cdot v + 45}{3^{2}}$

Step 2.3.1.8.5

Factor $3$ out of $3 v + 45$ .

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Step 2.3.1.8.5.1

Factor $3$ out of $3 v$ .

$2 v - 5 = \frac{3 (v) + 45}{3^{2}}$

Step 2.3.1.8.5.2

Factor $3$ out of $45$ .

$2 v - 5 = \frac{3 v + 3 \cdot 15}{3^{2}}$

Step 2.3.1.8.5.3

Factor $3$ out of $3 v + 3 \cdot 15$ .

$2 v - 5 = \frac{3 (v + 15)}{3^{2}}$

Step 2.3.1.9

Raise $3$ to the power of $2$ .

$2 v - 5 = \frac{3 (v + 15)}{9}$

Step 2.3.1.10

Cancel the common factors.

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Step 2.3.1.10.1

Factor $3$ out of $9$ .

$2 v - 5 = \frac{3 (v + 15)}{3 \cdot 3}$

Step 2.3.1.10.2

Cancel the common factor.

$2 v - 5 = \frac{3 (v + 15)}{3 \cdot 3}$

Step 2.3.1.10.3

Rewrite the expression.

$2 v - 5 = \frac{v + 15}{3}$

Step 3

Solve for $v$ .

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Step 3.1

Multiply both sides by $3$ .

$(2 v - 5) \cdot 3 = \frac{v + 15}{3} \cdot 3$

Step 3.2

Simplify.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $(2 v - 5) \cdot 3$ .

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Step 3.2.1.1.1

Apply the distributive property.

$2 v \cdot 3 - 5 \cdot 3 = \frac{v + 15}{3} \cdot 3$

Step 3.2.1.1.2

Multiply.

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Step 3.2.1.1.2.1

Multiply $3$ by $2$ .

$6 v - 5 \cdot 3 = \frac{v + 15}{3} \cdot 3$

Step 3.2.1.1.2.2

Multiply $- 5$ by $3$ .

$6 v - 15 = \frac{v + 15}{3} \cdot 3$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Cancel the common factor of $3$ .

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Step 3.2.2.1.1

Cancel the common factor.

$6 v - 15 = \frac{v + 15}{3} \cdot 3$

Step 3.2.2.1.2

Rewrite the expression.

$6 v - 15 = v + 15$

Step 3.3

Solve for $v$ .

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Step 3.3.1

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

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Step 3.3.1.1

Subtract $v$ from both sides of the equation.

$6 v - 15 - v = 15$

Step 3.3.1.2

Subtract $v$ from $6 v$ .

$5 v - 15 = 15$

Step 3.3.2

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

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Step 3.3.2.1

Add $15$ to both sides of the equation.

$5 v = 15 + 15$

Step 3.3.2.2

Add $15$ and $15$ .

$5 v = 30$

Step 3.3.3

Divide each term in $5 v = 30$ by $5$ and simplify.

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Step 3.3.3.1

Divide each term in $5 v = 30$ by $5$ .

$\frac{5 v}{5} = \frac{30}{5}$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $5$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{5 v}{5} = \frac{30}{5}$

Step 3.3.3.2.1.2

Divide $v$ by $1$ .

$v = \frac{30}{5}$

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Divide $30$ by $5$ .

$v = 6$