Solve for n square root of n+5- square root of n-10=1

$\sqrt{n + 5} - \sqrt{n - 10} = 1$

Step 1

Add $\sqrt{n - 10}$ to both sides of the equation.

$\sqrt{n + 5} = 1 + \sqrt{n - 10}$

Step 2

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

${\sqrt{n + 5}}^{2} = {(1 + \sqrt{n - 10})}^{2}$

Step 3

Simplify each side of the equation.

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

${({(n + 5)}^{\frac{1}{2}})}^{2} = {(1 + \sqrt{n - 10})}^{2}$

Simplify the left side.

Simplify ${({(n + 5)}^{\frac{1}{2}})}^{2}$ .

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

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

${(n + 5)}^{\frac{1}{2} \cdot 2} = {(1 + \sqrt{n - 10})}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(n + 5)}^{\frac{1}{2} \cdot 2} = {(1 + \sqrt{n - 10})}^{2}$

Rewrite the expression.

${(n + 5)}^{1} = {(1 + \sqrt{n - 10})}^{2}$

Simplify.

$n + 5 = {(1 + \sqrt{n - 10})}^{2}$

Simplify the right side.

Simplify ${(1 + \sqrt{n - 10})}^{2}$ .

Rewrite ${(1 + \sqrt{n - 10})}^{2}$ as $(1 + \sqrt{n - 10}) (1 + \sqrt{n - 10})$ .

$n + 5 = (1 + \sqrt{n - 10}) (1 + \sqrt{n - 10})$

Expand $(1 + \sqrt{n - 10}) (1 + \sqrt{n - 10})$ using the FOIL Method.

Apply the distributive property.

$n + 5 = 1 (1 + \sqrt{n - 10}) + \sqrt{n - 10} (1 + \sqrt{n - 10})$

Apply the distributive property.

$n + 5 = 1 \cdot 1 + 1 \sqrt{n - 10} + \sqrt{n - 10} (1 + \sqrt{n - 10})$

Apply the distributive property.

$n + 5 = 1 \cdot 1 + 1 \sqrt{n - 10} + \sqrt{n - 10} \cdot 1 + \sqrt{n - 10} \sqrt{n - 10}$

Simplify and combine like terms.

Simplify each term.

Multiply $1$ by $1$ .

$n + 5 = 1 + 1 \sqrt{n - 10} + \sqrt{n - 10} \cdot 1 + \sqrt{n - 10} \sqrt{n - 10}$

Multiply $\sqrt{n - 10}$ by $1$ .

$n + 5 = 1 + \sqrt{n - 10} + \sqrt{n - 10} \cdot 1 + \sqrt{n - 10} \sqrt{n - 10}$

Multiply $\sqrt{n - 10}$ by $1$ .

$n + 5 = 1 + \sqrt{n - 10} + \sqrt{n - 10} + \sqrt{n - 10} \sqrt{n - 10}$

Multiply $\sqrt{n - 10} \sqrt{n - 10}$ .

Raise $\sqrt{n - 10}$ to the power of $1$ .

$n + 5 = 1 + \sqrt{n - 10} + \sqrt{n - 10} + {\sqrt{n - 10}}^{1} \sqrt{n - 10}$

Raise $\sqrt{n - 10}$ to the power of $1$ .

$n + 5 = 1 + \sqrt{n - 10} + \sqrt{n - 10} + {\sqrt{n - 10}}^{1} {\sqrt{n - 10}}^{1}$

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

$n + 5 = 1 + \sqrt{n - 10} + \sqrt{n - 10} + {\sqrt{n - 10}}^{1 + 1}$

Add $1$ and $1$ .

$n + 5 = 1 + \sqrt{n - 10} + \sqrt{n - 10} + {\sqrt{n - 10}}^{2}$

Rewrite ${\sqrt{n - 10}}^{2}$ as $n - 10$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{n - 10}$ as ${(n - 10)}^{\frac{1}{2}}$ .

$n + 5 = 1 + \sqrt{n - 10} + \sqrt{n - 10} + {({(n - 10)}^{\frac{1}{2}})}^{2}$

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

$n + 5 = 1 + \sqrt{n - 10} + \sqrt{n - 10} + {(n - 10)}^{\frac{1}{2} \cdot 2}$

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

$n + 5 = 1 + \sqrt{n - 10} + \sqrt{n - 10} + {(n - 10)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$n + 5 = 1 + \sqrt{n - 10} + \sqrt{n - 10} + {(n - 10)}^{\frac{2}{2}}$

Rewrite the expression.

$n + 5 = 1 + \sqrt{n - 10} + \sqrt{n - 10} + {(n - 10)}^{1}$

Simplify.

$n + 5 = 1 + \sqrt{n - 10} + \sqrt{n - 10} + n - 10$

Subtract $10$ from $1$ .

$n + 5 = - 9 + \sqrt{n - 10} + \sqrt{n - 10} + n$

Add $\sqrt{n - 10}$ and $\sqrt{n - 10}$ .

$n + 5 = - 9 + 2 \sqrt{n - 10} + n$

Step 4

Solve for $2 \sqrt{n - 10}$ .

Rewrite the equation as $- 9 + 2 \sqrt{n - 10} + n = n + 5$ .

$- 9 + 2 \sqrt{n - 10} + n = n + 5$

Move all terms not containing $2 \sqrt{n - 10}$ to the right side of the equation.

Add $9$ to both sides of the equation.

$2 \sqrt{n - 10} + n = n + 5 + 9$

Subtract $n$ from both sides of the equation.

$2 \sqrt{n - 10} = n + 5 + 9 - n$

Combine the opposite terms in $n + 5 + 9 - n$ .

Subtract $n$ from $n$ .

$2 \sqrt{n - 10} = 0 + 5 + 9$

Add $0$ and $5$ .

$2 \sqrt{n - 10} = 5 + 9$

Add $5$ and $9$ .

$2 \sqrt{n - 10} = 14$

Step 5

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

${(2 \sqrt{n - 10})}^{2} = 14^{2}$

Step 6

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{n - 10}$ as ${(n - 10)}^{\frac{1}{2}}$ .

${(2 {(n - 10)}^{\frac{1}{2}})}^{2} = 14^{2}$

Simplify the left side.

Simplify ${(2 {(n - 10)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $2 {(n - 10)}^{\frac{1}{2}}$ .

$2^{2} {({(n - 10)}^{\frac{1}{2}})}^{2} = 14^{2}$

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

$4 {({(n - 10)}^{\frac{1}{2}})}^{2} = 14^{2}$

Multiply the exponents in ${({(n - 10)}^{\frac{1}{2}})}^{2}$ .

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

$4 {(n - 10)}^{\frac{1}{2} \cdot 2} = 14^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 {(n - 10)}^{\frac{1}{2} \cdot 2} = 14^{2}$

Rewrite the expression.

$4 {(n - 10)}^{1} = 14^{2}$

Simplify.

$4 (n - 10) = 14^{2}$

Apply the distributive property.

$4 n + 4 \cdot - 10 = 14^{2}$

Multiply $4$ by $- 10$ .

$4 n - 40 = 14^{2}$

Simplify the right side.

Raise $14$ to the power of $2$ .

$4 n - 40 = 196$

Step 7

Solve for $n$ .

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

Add $40$ to both sides of the equation.

$4 n = 196 + 40$

Add $196$ and $40$ .

$4 n = 236$

Divide each term in $4 n = 236$ by $4$ and simplify.

Divide each term in $4 n = 236$ by $4$ .

$\frac{4 n}{4} = \frac{236}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 n}{4} = \frac{236}{4}$

Divide $n$ by $1$ .

$n = \frac{236}{4}$

Simplify the right side.

Divide $236$ by $4$ .

$n = 59$

Solve for n square root of n+5- square root of n-10=1

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