Basic Math Examples

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

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.

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Step 2.1

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

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

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

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

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Step 2.2.1.1.1

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

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

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.

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

Step 2.2.1.1.2.2

Rewrite the expression.

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

Step 2.2.1.2

Simplify.

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

Step 2.3

Simplify the right side.

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Step 2.3.1

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

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Step 2.3.1.1

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

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

Step 2.3.1.2

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

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Step 2.3.1.2.1

Apply the distributive property.

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

Step 2.3.1.2.2

Apply the distributive property.

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

Step 2.3.1.2.3

Apply the distributive property.

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

Step 2.3.1.3

Simplify and combine like terms.

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Step 2.3.1.3.1

Simplify each term.

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Step 2.3.1.3.1.1

Multiply $1$ by $1$ .

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

Step 2.3.1.3.1.2

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

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

Step 2.3.1.3.1.3

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

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

Step 2.3.1.3.1.4

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

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Step 2.3.1.3.1.4.1

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

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

Step 2.3.1.3.1.4.2

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

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

Step 2.3.1.3.1.4.3

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

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

Step 2.3.1.3.1.4.4

Add $1$ and $1$ .

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

Step 2.3.1.3.1.5

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

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Step 2.3.1.3.1.5.1

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

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

Step 2.3.1.3.1.5.2

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

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

Step 2.3.1.3.1.5.3

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

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

Step 2.3.1.3.1.5.4

Cancel the common factor of $2$ .

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Step 2.3.1.3.1.5.4.1

Cancel the common factor.

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

Step 2.3.1.3.1.5.4.2

Rewrite the expression.

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

Step 2.3.1.3.1.5.5

Simplify.

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

Step 2.3.1.3.2

Subtract $9$ from $1$ .

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

Step 2.3.1.3.3

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

Add $8$ to both sides of the equation.

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

Step 3.2.2

Subtract $a$ from both sides of the equation.

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

Step 3.2.3

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

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Step 3.2.3.1

Subtract $a$ from $a$ .

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

Step 3.2.3.2

Add $0$ and $6$ .

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

Step 3.2.4

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.

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Step 5.1

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

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

Step 5.2

Simplify the left side.

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Step 5.2.1

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

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Step 5.2.1.1

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

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

Step 5.2.1.2

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

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

Step 5.2.1.3

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

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Step 5.2.1.3.1

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

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

Step 5.2.1.3.2

Cancel the common factor of $2$ .

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Step 5.2.1.3.2.1

Cancel the common factor.

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

Step 5.2.1.3.2.2

Rewrite the expression.

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

Step 5.2.1.4

Simplify.

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

Step 5.2.1.5

Apply the distributive property.

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

Step 5.2.1.6

Multiply $4$ by $- 9$ .

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

Step 5.3

Simplify the right side.

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Step 5.3.1

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

$4 a - 36 = 196$

Step 6

Solve for $a$ .

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Step 6.1

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

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Step 6.1.1

Add $36$ to both sides of the equation.

$4 a = 196 + 36$

Step 6.1.2

Add $196$ and $36$ .

$4 a = 232$

Step 6.2

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

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Step 6.2.1

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

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

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $4$ .

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Step 6.2.2.1.1

Cancel the common factor.

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

Step 6.2.2.1.2

Divide $a$ by $1$ .

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

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Divide $232$ by $4$ .

$a = 58$