Solve for v square root of 2v-5 = square root of v/3+5

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

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

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

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 v - 5}$ as ${(2 v - 5)}^{\frac{1}{2}}$ .

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

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

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

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

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

Combine the numerators over the common denominator.

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

Multiply $5$ by $3$ .

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

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

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

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

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

Simplify terms.

Combine and simplify the denominator.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Combine using the product rule for radicals.

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

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

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

Simplify the numerator.

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Apply the distributive property.

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

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

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

Multiply $15$ by $3$ .

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

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

Factor $3$ out of $3 v$ .

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

Factor $3$ out of $45$ .

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

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

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

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

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

Cancel the common factors.

Factor $3$ out of $9$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Solve for $v$ .

Multiply both sides by $3$ .

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

Simplify.

Simplify the left side.

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

Apply the distributive property.

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

Multiply.

Multiply $3$ by $2$ .

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

Multiply $- 5$ by $3$ .

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

Simplify the right side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

$6 v - 15 = v + 15$

Solve for $v$ .

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

Subtract $v$ from both sides of the equation.

$6 v - 15 - v = 15$

Subtract $v$ from $6 v$ .

$5 v - 15 = 15$

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

Add $15$ to both sides of the equation.

$5 v = 15 + 15$

Add $15$ and $15$ .

$5 v = 30$

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

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

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

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

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

Divide $v$ by $1$ .

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

Simplify the right side.

Divide $30$ by $5$ .

$v = 6$

Solve for v square root of 2v-5 = square root of v/3+5

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