Solve for n square root of n-15=3- square root of n

$\sqrt{n - 15} = 3 - \sqrt{n}$

Step 1

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

${\sqrt{n - 15}}^{2} = {(3 - \sqrt{n})}^{2}$

Step 2

Simplify each side of the equation.

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

${({(n - 15)}^{\frac{1}{2}})}^{2} = {(3 - \sqrt{n})}^{2}$

Simplify the left side.

Simplify ${({(n - 15)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(n - 15)}^{\frac{1}{2}})}^{2}$ .

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

${(n - 15)}^{1} = {(3 - \sqrt{n})}^{2}$

Simplify.

$n - 15 = {(3 - \sqrt{n})}^{2}$

Simplify the right side.

Simplify ${(3 - \sqrt{n})}^{2}$ .

Rewrite ${(3 - \sqrt{n})}^{2}$ as $(3 - \sqrt{n}) (3 - \sqrt{n})$ .

$n - 15 = (3 - \sqrt{n}) (3 - \sqrt{n})$

Expand $(3 - \sqrt{n}) (3 - \sqrt{n})$ using the FOIL Method.

Apply the distributive property.

$n - 15 = 3 (3 - \sqrt{n}) - \sqrt{n} (3 - \sqrt{n})$

Apply the distributive property.

$n - 15 = 3 \cdot 3 + 3 (- \sqrt{n}) - \sqrt{n} (3 - \sqrt{n})$

Apply the distributive property.

$n - 15 = 3 \cdot 3 + 3 (- \sqrt{n}) - \sqrt{n} \cdot 3 - \sqrt{n} (- \sqrt{n})$

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $3$ .

$n - 15 = 9 + 3 (- \sqrt{n}) - \sqrt{n} \cdot 3 - \sqrt{n} (- \sqrt{n})$

Multiply $- 1$ by $3$ .

$n - 15 = 9 - 3 \sqrt{n} - \sqrt{n} \cdot 3 - \sqrt{n} (- \sqrt{n})$

Multiply $3$ by $- 1$ .

$n - 15 = 9 - 3 \sqrt{n} - 3 \sqrt{n} - \sqrt{n} (- \sqrt{n})$

Multiply $- \sqrt{n} (- \sqrt{n})$ .

Multiply $- 1$ by $- 1$ .

$n - 15 = 9 - 3 \sqrt{n} - 3 \sqrt{n} + 1 \sqrt{n} \sqrt{n}$

Multiply $\sqrt{n}$ by $1$ .

$n - 15 = 9 - 3 \sqrt{n} - 3 \sqrt{n} + \sqrt{n} \sqrt{n}$

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

$n - 15 = 9 - 3 \sqrt{n} - 3 \sqrt{n} + {\sqrt{n}}^{1} \sqrt{n}$

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

$n - 15 = 9 - 3 \sqrt{n} - 3 \sqrt{n} + {\sqrt{n}}^{1} {\sqrt{n}}^{1}$

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

$n - 15 = 9 - 3 \sqrt{n} - 3 \sqrt{n} + {\sqrt{n}}^{1 + 1}$

Add $1$ and $1$ .

$n - 15 = 9 - 3 \sqrt{n} - 3 \sqrt{n} + {\sqrt{n}}^{2}$

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

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

$n - 15 = 9 - 3 \sqrt{n} - 3 \sqrt{n} + {(n^{\frac{1}{2}})}^{2}$

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

$n - 15 = 9 - 3 \sqrt{n} - 3 \sqrt{n} + n^{\frac{1}{2} \cdot 2}$

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

$n - 15 = 9 - 3 \sqrt{n} - 3 \sqrt{n} + n^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$n - 15 = 9 - 3 \sqrt{n} - 3 \sqrt{n} + n^{\frac{2}{2}}$

Rewrite the expression.

$n - 15 = 9 - 3 \sqrt{n} - 3 \sqrt{n} + n^{1}$

Simplify.

$n - 15 = 9 - 3 \sqrt{n} - 3 \sqrt{n} + n$

Subtract $3 \sqrt{n}$ from $- 3 \sqrt{n}$ .

$n - 15 = 9 - 6 \sqrt{n} + n$

Step 3

Solve for $- 6 \sqrt{n}$ .

Rewrite the equation as $9 - 6 \sqrt{n} + n = n - 15$ .

$9 - 6 \sqrt{n} + n = n - 15$

Move all terms not containing $- 6 \sqrt{n}$ to the right side of the equation.

Subtract $9$ from both sides of the equation.

$- 6 \sqrt{n} + n = n - 15 - 9$

Subtract $n$ from both sides of the equation.

$- 6 \sqrt{n} = n - 15 - 9 - n$

Combine the opposite terms in $n - 15 - 9 - n$ .

Subtract $n$ from $n$ .

$- 6 \sqrt{n} = 0 - 15 - 9$

Subtract $15$ from $0$ .

$- 6 \sqrt{n} = - 15 - 9$

Subtract $9$ from $- 15$ .

$- 6 \sqrt{n} = - 24$

Step 4

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

${(- 6 \sqrt{n})}^{2} = {(- 24)}^{2}$

Step 5

Simplify each side of the equation.

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

${(- 6 n^{\frac{1}{2}})}^{2} = {(- 24)}^{2}$

Simplify the left side.

Simplify ${(- 6 n^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $- 6 n^{\frac{1}{2}}$ .

${(- 6)}^{2} {(n^{\frac{1}{2}})}^{2} = {(- 24)}^{2}$

Raise $- 6$ to the power of $2$ .

$36 {(n^{\frac{1}{2}})}^{2} = {(- 24)}^{2}$

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

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

$36 n^{\frac{1}{2} \cdot 2} = {(- 24)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$36 n^{\frac{1}{2} \cdot 2} = {(- 24)}^{2}$

Rewrite the expression.

$36 n^{1} = {(- 24)}^{2}$

Simplify.

$36 n = {(- 24)}^{2}$

Simplify the right side.

Raise $- 24$ to the power of $2$ .

$36 n = 576$

Step 6

Divide each term in $36 n = 576$ by $36$ and simplify.

Divide each term in $36 n = 576$ by $36$ .

$\frac{36 n}{36} = \frac{576}{36}$

Simplify the left side.

Cancel the common factor of $36$ .

Cancel the common factor.

$\frac{36 n}{36} = \frac{576}{36}$

Divide $n$ by $1$ .

$n = \frac{576}{36}$

Simplify the right side.

Divide $576$ by $36$ .

$n = 16$

Step 7

Exclude the solutions that do not make $\sqrt{n - 15} = 3 - \sqrt{n}$ true.

$No solution$

Solve for n square root of n-15=3- square root of n

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