Solve for a square root of a+6=1+ square root of a-9

$\sqrt{a + 6} = 1 + \sqrt{a - 9}$

Step 1

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

${\sqrt{a + 6}}^{2} = {(1 + \sqrt{a - 9})}^{2}$

Step 2

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{a + 6}$ as ${(a + 6)}^{\frac{1}{2}}$ .

${({(a + 6)}^{\frac{1}{2}})}^{2} = {(1 + \sqrt{a - 9})}^{2}$

Simplify the left side.

Simplify ${({(a + 6)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(a + 6)}^{\frac{1}{2}})}^{2}$ .

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

${(a + 6)}^{\frac{1}{2} \cdot 2} = {(1 + \sqrt{a - 9})}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(a + 6)}^{\frac{1}{2} \cdot 2} = {(1 + \sqrt{a - 9})}^{2}$

Rewrite the expression.

${(a + 6)}^{1} = {(1 + \sqrt{a - 9})}^{2}$

Simplify.

$a + 6 = {(1 + \sqrt{a - 9})}^{2}$

Simplify the right side.

Simplify ${(1 + \sqrt{a - 9})}^{2}$ .

Rewrite ${(1 + \sqrt{a - 9})}^{2}$ as $(1 + \sqrt{a - 9}) (1 + \sqrt{a - 9})$ .

$a + 6 = (1 + \sqrt{a - 9}) (1 + \sqrt{a - 9})$

Expand $(1 + \sqrt{a - 9}) (1 + \sqrt{a - 9})$ using the FOIL Method.

Apply the distributive property.

$a + 6 = 1 (1 + \sqrt{a - 9}) + \sqrt{a - 9} (1 + \sqrt{a - 9})$

Apply the distributive property.

$a + 6 = 1 \cdot 1 + 1 \sqrt{a - 9} + \sqrt{a - 9} (1 + \sqrt{a - 9})$

Apply the distributive property.

$a + 6 = 1 \cdot 1 + 1 \sqrt{a - 9} + \sqrt{a - 9} \cdot 1 + \sqrt{a - 9} \sqrt{a - 9}$

Simplify and combine like terms.

Simplify each term.

Multiply $1$ by $1$ .

$a + 6 = 1 + 1 \sqrt{a - 9} + \sqrt{a - 9} \cdot 1 + \sqrt{a - 9} \sqrt{a - 9}$

Multiply $\sqrt{a - 9}$ by $1$ .

$a + 6 = 1 + \sqrt{a - 9} + \sqrt{a - 9} \cdot 1 + \sqrt{a - 9} \sqrt{a - 9}$

Multiply $\sqrt{a - 9}$ by $1$ .

$a + 6 = 1 + \sqrt{a - 9} + \sqrt{a - 9} + \sqrt{a - 9} \sqrt{a - 9}$

Multiply $\sqrt{a - 9} \sqrt{a - 9}$ .

Raise $\sqrt{a - 9}$ to the power of $1$ .

$a + 6 = 1 + \sqrt{a - 9} + \sqrt{a - 9} + {\sqrt{a - 9}}^{1} \sqrt{a - 9}$

Raise $\sqrt{a - 9}$ to the power of $1$ .

$a + 6 = 1 + \sqrt{a - 9} + \sqrt{a - 9} + {\sqrt{a - 9}}^{1} {\sqrt{a - 9}}^{1}$

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

$a + 6 = 1 + \sqrt{a - 9} + \sqrt{a - 9} + {\sqrt{a - 9}}^{1 + 1}$

Add $1$ and $1$ .

$a + 6 = 1 + \sqrt{a - 9} + \sqrt{a - 9} + {\sqrt{a - 9}}^{2}$

Rewrite ${\sqrt{a - 9}}^{2}$ as $a - 9$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{a - 9}$ as ${(a - 9)}^{\frac{1}{2}}$ .

$a + 6 = 1 + \sqrt{a - 9} + \sqrt{a - 9} + {({(a - 9)}^{\frac{1}{2}})}^{2}$

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

$a + 6 = 1 + \sqrt{a - 9} + \sqrt{a - 9} + {(a - 9)}^{\frac{1}{2} \cdot 2}$

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

$a + 6 = 1 + \sqrt{a - 9} + \sqrt{a - 9} + {(a - 9)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$a + 6 = 1 + \sqrt{a - 9} + \sqrt{a - 9} + {(a - 9)}^{\frac{2}{2}}$

Rewrite the expression.

$a + 6 = 1 + \sqrt{a - 9} + \sqrt{a - 9} + {(a - 9)}^{1}$

Simplify.

$a + 6 = 1 + \sqrt{a - 9} + \sqrt{a - 9} + a - 9$

Subtract $9$ from $1$ .

$a + 6 = - 8 + \sqrt{a - 9} + \sqrt{a - 9} + a$

Add $\sqrt{a - 9}$ and $\sqrt{a - 9}$ .

$a + 6 = - 8 + 2 \sqrt{a - 9} + a$

Step 3

Solve for $2 \sqrt{a - 9}$ .

Rewrite the equation as $- 8 + 2 \sqrt{a - 9} + a = a + 6$ .

$- 8 + 2 \sqrt{a - 9} + a = a + 6$

Move all terms not containing $2 \sqrt{a - 9}$ to the right side of the equation.

Add $8$ to both sides of the equation.

$2 \sqrt{a - 9} + a = a + 6 + 8$

Subtract $a$ from both sides of the equation.

$2 \sqrt{a - 9} = a + 6 + 8 - a$

Combine the opposite terms in $a + 6 + 8 - a$ .

Subtract $a$ from $a$ .

$2 \sqrt{a - 9} = 0 + 6 + 8$

Add $0$ and $6$ .

$2 \sqrt{a - 9} = 6 + 8$

Add $6$ and $8$ .

$2 \sqrt{a - 9} = 14$

Step 4

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

${(2 \sqrt{a - 9})}^{2} = 14^{2}$

Step 5

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{a - 9}$ as ${(a - 9)}^{\frac{1}{2}}$ .

${(2 {(a - 9)}^{\frac{1}{2}})}^{2} = 14^{2}$

Simplify the left side.

Simplify ${(2 {(a - 9)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $2 {(a - 9)}^{\frac{1}{2}}$ .

$2^{2} {({(a - 9)}^{\frac{1}{2}})}^{2} = 14^{2}$

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

$4 {({(a - 9)}^{\frac{1}{2}})}^{2} = 14^{2}$

Multiply the exponents in ${({(a - 9)}^{\frac{1}{2}})}^{2}$ .

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

$4 {(a - 9)}^{\frac{1}{2} \cdot 2} = 14^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 {(a - 9)}^{\frac{1}{2} \cdot 2} = 14^{2}$

Rewrite the expression.

$4 {(a - 9)}^{1} = 14^{2}$

Simplify.

$4 (a - 9) = 14^{2}$

Apply the distributive property.

$4 a + 4 \cdot - 9 = 14^{2}$

Multiply $4$ by $- 9$ .

$4 a - 36 = 14^{2}$

Simplify the right side.

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

$4 a - 36 = 196$

Step 6

Solve for $a$ .

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

Add $36$ to both sides of the equation.

$4 a = 196 + 36$

Add $196$ and $36$ .

$4 a = 232$

Divide each term in $4 a = 232$ by $4$ and simplify.

Divide each term in $4 a = 232$ by $4$ .

$\frac{4 a}{4} = \frac{232}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 a}{4} = \frac{232}{4}$

Divide $a$ by $1$ .

$a = \frac{232}{4}$

Simplify the right side.

Divide $232$ by $4$ .

$a = 58$

Solve for a square root of a+6=1+ square root of a-9

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