Solve the Rational Equation for n square root of 3n = square root of 4n-1

$\sqrt{3 n} = \sqrt{4 n - 1}$

Step 1

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

${\sqrt{3 n}}^{2} = {\sqrt{4 n - 1}}^{2}$

Step 2

Simplify each side of the equation.

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

${({(3 n)}^{\frac{1}{2}})}^{2} = {\sqrt{4 n - 1}}^{2}$

Simplify the left side.

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

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

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

${(3 n)}^{\frac{1}{2} \cdot 2} = {\sqrt{4 n - 1}}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(3 n)}^{\frac{1}{2} \cdot 2} = {\sqrt{4 n - 1}}^{2}$

Rewrite the expression.

${(3 n)}^{1} = {\sqrt{4 n - 1}}^{2}$

Simplify.

$3 n = {\sqrt{4 n - 1}}^{2}$

Simplify the right side.

Rewrite ${\sqrt{4 n - 1}}^{2}$ as $4 n - 1$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 n - 1}$ as ${(4 n - 1)}^{\frac{1}{2}}$ .

$3 n = {({(4 n - 1)}^{\frac{1}{2}})}^{2}$

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

$3 n = {(4 n - 1)}^{\frac{1}{2} \cdot 2}$

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

$3 n = {(4 n - 1)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$3 n = {(4 n - 1)}^{\frac{2}{2}}$

Rewrite the expression.

$3 n = {(4 n - 1)}^{1}$

Simplify.

$3 n = 4 n - 1$

Step 3

Solve for $n$ .

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

Subtract $4 n$ from both sides of the equation.

$3 n - 4 n = - 1$

Subtract $4 n$ from $3 n$ .

$- n = - 1$

Divide each term in $- n = - 1$ by $- 1$ and simplify.

Divide each term in $- n = - 1$ by $- 1$ .

$\frac{- n}{- 1} = \frac{- 1}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{n}{1} = \frac{- 1}{- 1}$

Divide $n$ by $1$ .

$n = \frac{- 1}{- 1}$

Simplify the right side.

Divide $- 1$ by $- 1$ .

$n = 1$

Solve the Rational Equation for n square root of 3n = square root of 4n-1

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