Precalculus Examples

$\sqrt{3 n} = n$

Step 1

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

${\sqrt{3 n}}^{2} = n^{2}$

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

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

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

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

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Step 2.2.1.1.1

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

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

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.

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

Step 2.2.1.1.2.2

Rewrite the expression.

${(3 n)}^{1} = n^{2}$

Step 2.2.1.2

Simplify.

$3 n = n^{2}$

Step 3

Solve for $n$ .

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Step 3.1

Subtract $n^{2}$ from both sides of the equation.

$3 n - n^{2} = 0$

Step 3.2

Factor the left side of the equation.

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Step 3.2.1

Let $u = n$ . Substitute $u$ for all occurrences of $n$ .

$3 u - u^{2} = 0$

Step 3.2.2

Factor $u$ out of $3 u - u^{2}$ .

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Step 3.2.2.1

Factor $u$ out of $3 u$ .

$u \cdot 3 - u^{2} = 0$

Step 3.2.2.2

Factor $u$ out of $- u^{2}$ .

$u \cdot 3 + u (- u) = 0$

Step 3.2.2.3

Factor $u$ out of $u \cdot 3 + u (- u)$ .

$u (3 - u) = 0$

Step 3.2.3

Replace all occurrences of $u$ with $n$ .

$n (3 - n) = 0$

Step 3.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$n = 0$

$3 - n = 0$

Step 3.4

Set $n$ equal to $0$ .

$n = 0$

Step 3.5

Set $3 - n$ equal to $0$ and solve for $n$ .

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Step 3.5.1

Set $3 - n$ equal to $0$ .

$3 - n = 0$

Step 3.5.2

Solve $3 - n = 0$ for $n$ .

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Step 3.5.2.1

Subtract $3$ from both sides of the equation.

$- n = - 3$

Step 3.5.2.2

Divide each term in $- n = - 3$ by $- 1$ and simplify.

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Step 3.5.2.2.1

Divide each term in $- n = - 3$ by $- 1$ .

$\frac{- n}{- 1} = \frac{- 3}{- 1}$

Step 3.5.2.2.2

Simplify the left side.

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Step 3.5.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{n}{1} = \frac{- 3}{- 1}$

Step 3.5.2.2.2.2

Divide $n$ by $1$ .

$n = \frac{- 3}{- 1}$

Step 3.5.2.2.3

Simplify the right side.

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Step 3.5.2.2.3.1

Divide $- 3$ by $- 1$ .

$n = 3$

Step 3.6

The final solution is all the values that make $n (3 - n) = 0$ true.

$n = 0, 3$