Finite Math Examples

$\sqrt{z - 7} + \sqrt{z + 9} = 8$

Step 1

Subtract $\sqrt{z + 9}$ from both sides of the equation.

$\sqrt{z - 7} = 8 - \sqrt{z + 9}$

Step 2

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

${\sqrt{z - 7}}^{2} = {(8 - \sqrt{z + 9})}^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{z - 7}$ as ${(z - 7)}^{\frac{1}{2}}$ .

${({(z - 7)}^{\frac{1}{2}})}^{2} = {(8 - \sqrt{z + 9})}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(z - 7)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(z - 7)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(z - 7)}^{\frac{1}{2} \cdot 2} = {(8 - \sqrt{z + 9})}^{2}$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

${(z - 7)}^{\frac{1}{2} \cdot 2} = {(8 - \sqrt{z + 9})}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(z - 7)}^{1} = {(8 - \sqrt{z + 9})}^{2}$

Step 3.2.1.2

Simplify.

$z - 7 = {(8 - \sqrt{z + 9})}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(8 - \sqrt{z + 9})}^{2}$ .

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Step 3.3.1.1

Rewrite ${(8 - \sqrt{z + 9})}^{2}$ as $(8 - \sqrt{z + 9}) (8 - \sqrt{z + 9})$ .

$z - 7 = (8 - \sqrt{z + 9}) (8 - \sqrt{z + 9})$

Step 3.3.1.2

Expand $(8 - \sqrt{z + 9}) (8 - \sqrt{z + 9})$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

$z - 7 = 8 (8 - \sqrt{z + 9}) - \sqrt{z + 9} (8 - \sqrt{z + 9})$

Step 3.3.1.2.2

Apply the distributive property.

$z - 7 = 8 \cdot 8 + 8 (- \sqrt{z + 9}) - \sqrt{z + 9} (8 - \sqrt{z + 9})$

Step 3.3.1.2.3

Apply the distributive property.

$z - 7 = 8 \cdot 8 + 8 (- \sqrt{z + 9}) - \sqrt{z + 9} \cdot 8 - \sqrt{z + 9} (- \sqrt{z + 9})$

Step 3.3.1.3

Simplify and combine like terms.

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Step 3.3.1.3.1

Simplify each term.

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Step 3.3.1.3.1.1

Multiply $8$ by $8$ .

$z - 7 = 64 + 8 (- \sqrt{z + 9}) - \sqrt{z + 9} \cdot 8 - \sqrt{z + 9} (- \sqrt{z + 9})$

Step 3.3.1.3.1.2

Multiply $- 1$ by $8$ .

$z - 7 = 64 - 8 \sqrt{z + 9} - \sqrt{z + 9} \cdot 8 - \sqrt{z + 9} (- \sqrt{z + 9})$

Step 3.3.1.3.1.3

Multiply $8$ by $- 1$ .

$z - 7 = 64 - 8 \sqrt{z + 9} - 8 \sqrt{z + 9} - \sqrt{z + 9} (- \sqrt{z + 9})$

Step 3.3.1.3.1.4

Multiply $- \sqrt{z + 9} (- \sqrt{z + 9})$ .

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Step 3.3.1.3.1.4.1

Multiply $- 1$ by $- 1$ .

$z - 7 = 64 - 8 \sqrt{z + 9} - 8 \sqrt{z + 9} + 1 \sqrt{z + 9} \sqrt{z + 9}$

Step 3.3.1.3.1.4.2

Multiply $\sqrt{z + 9}$ by $1$ .

$z - 7 = 64 - 8 \sqrt{z + 9} - 8 \sqrt{z + 9} + \sqrt{z + 9} \sqrt{z + 9}$

Step 3.3.1.3.1.4.3

Raise $\sqrt{z + 9}$ to the power of $1$ .

$z - 7 = 64 - 8 \sqrt{z + 9} - 8 \sqrt{z + 9} + {\sqrt{z + 9}}^{1} \sqrt{z + 9}$

Step 3.3.1.3.1.4.4

Raise $\sqrt{z + 9}$ to the power of $1$ .

$z - 7 = 64 - 8 \sqrt{z + 9} - 8 \sqrt{z + 9} + {\sqrt{z + 9}}^{1} {\sqrt{z + 9}}^{1}$

Step 3.3.1.3.1.4.5

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

$z - 7 = 64 - 8 \sqrt{z + 9} - 8 \sqrt{z + 9} + {\sqrt{z + 9}}^{1 + 1}$

Step 3.3.1.3.1.4.6

Add $1$ and $1$ .

$z - 7 = 64 - 8 \sqrt{z + 9} - 8 \sqrt{z + 9} + {\sqrt{z + 9}}^{2}$

Step 3.3.1.3.1.5

Rewrite ${\sqrt{z + 9}}^{2}$ as $z + 9$ .

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Step 3.3.1.3.1.5.1

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

$z - 7 = 64 - 8 \sqrt{z + 9} - 8 \sqrt{z + 9} + {({(z + 9)}^{\frac{1}{2}})}^{2}$

Step 3.3.1.3.1.5.2

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

$z - 7 = 64 - 8 \sqrt{z + 9} - 8 \sqrt{z + 9} + {(z + 9)}^{\frac{1}{2} \cdot 2}$

Step 3.3.1.3.1.5.3

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

$z - 7 = 64 - 8 \sqrt{z + 9} - 8 \sqrt{z + 9} + {(z + 9)}^{\frac{2}{2}}$

Step 3.3.1.3.1.5.4

Cancel the common factor of $2$ .

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Step 3.3.1.3.1.5.4.1

Cancel the common factor.

$z - 7 = 64 - 8 \sqrt{z + 9} - 8 \sqrt{z + 9} + {(z + 9)}^{\frac{2}{2}}$

Step 3.3.1.3.1.5.4.2

Rewrite the expression.

$z - 7 = 64 - 8 \sqrt{z + 9} - 8 \sqrt{z + 9} + {(z + 9)}^{1}$

Step 3.3.1.3.1.5.5

Simplify.

$z - 7 = 64 - 8 \sqrt{z + 9} - 8 \sqrt{z + 9} + z + 9$

Step 3.3.1.3.2

Add $64$ and $9$ .

$z - 7 = 73 - 8 \sqrt{z + 9} - 8 \sqrt{z + 9} + z$

Step 3.3.1.3.3

Subtract $8 \sqrt{z + 9}$ from $- 8 \sqrt{z + 9}$ .

$z - 7 = 73 - 16 \sqrt{z + 9} + z$

Step 4

Solve for $- 16 \sqrt{z + 9}$ .

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Step 4.1

Rewrite the equation as $73 - 16 \sqrt{z + 9} + z = z - 7$ .

$73 - 16 \sqrt{z + 9} + z = z - 7$

Step 4.2

Move all terms not containing $- 16 \sqrt{z + 9}$ to the right side of the equation.

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Step 4.2.1

Subtract $73$ from both sides of the equation.

$- 16 \sqrt{z + 9} + z = z - 7 - 73$

Step 4.2.2

Subtract $z$ from both sides of the equation.

$- 16 \sqrt{z + 9} = z - 7 - 73 - z$

Step 4.2.3

Combine the opposite terms in $z - 7 - 73 - z$ .

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Step 4.2.3.1

Subtract $z$ from $z$ .

$- 16 \sqrt{z + 9} = 0 - 7 - 73$

Step 4.2.3.2

Subtract $7$ from $0$ .

$- 16 \sqrt{z + 9} = - 7 - 73$

Step 4.2.4

Subtract $73$ from $- 7$ .

$- 16 \sqrt{z + 9} = - 80$

Step 5

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

${(- 16 \sqrt{z + 9})}^{2} = {(- 80)}^{2}$

Step 6

Simplify each side of the equation.

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Step 6.1

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

${(- 16 {(z + 9)}^{\frac{1}{2}})}^{2} = {(- 80)}^{2}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Simplify ${(- 16 {(z + 9)}^{\frac{1}{2}})}^{2}$ .

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Step 6.2.1.1

Apply the product rule to $- 16 {(z + 9)}^{\frac{1}{2}}$ .

${(- 16)}^{2} {({(z + 9)}^{\frac{1}{2}})}^{2} = {(- 80)}^{2}$

Step 6.2.1.2

Raise $- 16$ to the power of $2$ .

$256 {({(z + 9)}^{\frac{1}{2}})}^{2} = {(- 80)}^{2}$

Step 6.2.1.3

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

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Step 6.2.1.3.1

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

$256 {(z + 9)}^{\frac{1}{2} \cdot 2} = {(- 80)}^{2}$

Step 6.2.1.3.2

Cancel the common factor of $2$ .

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Step 6.2.1.3.2.1

Cancel the common factor.

$256 {(z + 9)}^{\frac{1}{2} \cdot 2} = {(- 80)}^{2}$

Step 6.2.1.3.2.2

Rewrite the expression.

$256 {(z + 9)}^{1} = {(- 80)}^{2}$

Step 6.2.1.4

Simplify.

$256 (z + 9) = {(- 80)}^{2}$

Step 6.2.1.5

Apply the distributive property.

$256 z + 256 \cdot 9 = {(- 80)}^{2}$

Step 6.2.1.6

Multiply $256$ by $9$ .

$256 z + 2304 = {(- 80)}^{2}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Raise $- 80$ to the power of $2$ .

$256 z + 2304 = 6400$

Step 7

Solve for $z$ .

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Step 7.1

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

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Step 7.1.1

Subtract $2304$ from both sides of the equation.

$256 z = 6400 - 2304$

Step 7.1.2

Subtract $2304$ from $6400$ .

$256 z = 4096$

Step 7.2

Divide each term in $256 z = 4096$ by $256$ and simplify.

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Step 7.2.1

Divide each term in $256 z = 4096$ by $256$ .

$\frac{256 z}{256} = \frac{4096}{256}$

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

Cancel the common factor of $256$ .

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Step 7.2.2.1.1

Cancel the common factor.

$\frac{256 z}{256} = \frac{4096}{256}$

Step 7.2.2.1.2

Divide $z$ by $1$ .

$z = \frac{4096}{256}$

Step 7.2.3

Simplify the right side.

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Step 7.2.3.1

Divide $4096$ by $256$ .

$z = 16$