Finite Math Examples

$\sqrt{y + 40} = \sqrt{y} + 4$

Step 1

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

${\sqrt{y + 40}}^{2} = {(\sqrt{y} + 4)}^{2}$

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 $\sqrt{y + 40}$ as ${(y + 40)}^{\frac{1}{2}}$ .

${({(y + 40)}^{\frac{1}{2}})}^{2} = {(\sqrt{y} + 4)}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(y + 40)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(y + 40)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(y + 40)}^{\frac{1}{2} \cdot 2} = {(\sqrt{y} + 4)}^{2}$

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.

${(y + 40)}^{\frac{1}{2} \cdot 2} = {(\sqrt{y} + 4)}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(y + 40)}^{1} = {(\sqrt{y} + 4)}^{2}$

Step 2.2.1.2

Simplify.

$y + 40 = {(\sqrt{y} + 4)}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${(\sqrt{y} + 4)}^{2}$ .

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Step 2.3.1.1

Rewrite ${(\sqrt{y} + 4)}^{2}$ as $(\sqrt{y} + 4) (\sqrt{y} + 4)$ .

$y + 40 = (\sqrt{y} + 4) (\sqrt{y} + 4)$

Step 2.3.1.2

Expand $(\sqrt{y} + 4) (\sqrt{y} + 4)$ using the FOIL Method.

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Step 2.3.1.2.1

Apply the distributive property.

$y + 40 = \sqrt{y} (\sqrt{y} + 4) + 4 (\sqrt{y} + 4)$

Step 2.3.1.2.2

Apply the distributive property.

$y + 40 = \sqrt{y} \sqrt{y} + \sqrt{y} \cdot 4 + 4 (\sqrt{y} + 4)$

Step 2.3.1.2.3

Apply the distributive property.

$y + 40 = \sqrt{y} \sqrt{y} + \sqrt{y} \cdot 4 + 4 \sqrt{y} + 4 \cdot 4$

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 $\sqrt{y} \sqrt{y}$ .

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Step 2.3.1.3.1.1.1

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

$y + 40 = {\sqrt{y}}^{1} \sqrt{y} + \sqrt{y} \cdot 4 + 4 \sqrt{y} + 4 \cdot 4$

Step 2.3.1.3.1.1.2

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

$y + 40 = {\sqrt{y}}^{1} {\sqrt{y}}^{1} + \sqrt{y} \cdot 4 + 4 \sqrt{y} + 4 \cdot 4$

Step 2.3.1.3.1.1.3

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

$y + 40 = {\sqrt{y}}^{1 + 1} + \sqrt{y} \cdot 4 + 4 \sqrt{y} + 4 \cdot 4$

Step 2.3.1.3.1.1.4

Add $1$ and $1$ .

$y + 40 = {\sqrt{y}}^{2} + \sqrt{y} \cdot 4 + 4 \sqrt{y} + 4 \cdot 4$

Step 2.3.1.3.1.2

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

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Step 2.3.1.3.1.2.1

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

$y + 40 = {(y^{\frac{1}{2}})}^{2} + \sqrt{y} \cdot 4 + 4 \sqrt{y} + 4 \cdot 4$

Step 2.3.1.3.1.2.2

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

$y + 40 = y^{\frac{1}{2} \cdot 2} + \sqrt{y} \cdot 4 + 4 \sqrt{y} + 4 \cdot 4$

Step 2.3.1.3.1.2.3

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

$y + 40 = y^{\frac{2}{2}} + \sqrt{y} \cdot 4 + 4 \sqrt{y} + 4 \cdot 4$

Step 2.3.1.3.1.2.4

Cancel the common factor of $2$ .

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Step 2.3.1.3.1.2.4.1

Cancel the common factor.

$y + 40 = y^{\frac{2}{2}} + \sqrt{y} \cdot 4 + 4 \sqrt{y} + 4 \cdot 4$

Step 2.3.1.3.1.2.4.2

Rewrite the expression.

$y + 40 = y^{1} + \sqrt{y} \cdot 4 + 4 \sqrt{y} + 4 \cdot 4$

Step 2.3.1.3.1.2.5

Simplify.

$y + 40 = y + \sqrt{y} \cdot 4 + 4 \sqrt{y} + 4 \cdot 4$

Step 2.3.1.3.1.3

Move $4$ to the left of $\sqrt{y}$ .

$y + 40 = y + 4 \cdot \sqrt{y} + 4 \sqrt{y} + 4 \cdot 4$

Step 2.3.1.3.1.4

Multiply $4$ by $4$ .

$y + 40 = y + 4 \sqrt{y} + 4 \sqrt{y} + 16$

Step 2.3.1.3.2

Add $4 \sqrt{y}$ and $4 \sqrt{y}$ .

$y + 40 = y + 8 \sqrt{y} + 16$

Step 3

Solve for $8 \sqrt{y}$ .

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Step 3.1

Rewrite the equation as $y + 8 \sqrt{y} + 16 = y + 40$ .

$y + 8 \sqrt{y} + 16 = y + 40$

Step 3.2

Move all terms not containing $8 \sqrt{y}$ to the right side of the equation.

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Step 3.2.1

Subtract $y$ from both sides of the equation.

$8 \sqrt{y} + 16 = y + 40 - y$

Step 3.2.2

Subtract $16$ from both sides of the equation.

$8 \sqrt{y} = y + 40 - y - 16$

Step 3.2.3

Combine the opposite terms in $y + 40 - y - 16$ .

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Step 3.2.3.1

Subtract $y$ from $y$ .

$8 \sqrt{y} = 0 + 40 - 16$

Step 3.2.3.2

Add $0$ and $40$ .

$8 \sqrt{y} = 40 - 16$

Step 3.2.4

Subtract $16$ from $40$ .

$8 \sqrt{y} = 24$

Step 4

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

${(8 \sqrt{y})}^{2} = 24^{2}$

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 $\sqrt{y}$ as $y^{\frac{1}{2}}$ .

${(8 y^{\frac{1}{2}})}^{2} = 24^{2}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Simplify ${(8 y^{\frac{1}{2}})}^{2}$ .

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Step 5.2.1.1

Apply the product rule to $8 y^{\frac{1}{2}}$ .

$8^{2} {(y^{\frac{1}{2}})}^{2} = 24^{2}$

Step 5.2.1.2

Raise $8$ to the power of $2$ .

$64 {(y^{\frac{1}{2}})}^{2} = 24^{2}$

Step 5.2.1.3

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

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Step 5.2.1.3.1

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

$64 y^{\frac{1}{2} \cdot 2} = 24^{2}$

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.

$64 y^{\frac{1}{2} \cdot 2} = 24^{2}$

Step 5.2.1.3.2.2

Rewrite the expression.

$64 y^{1} = 24^{2}$

Step 5.2.1.4

Simplify.

$64 y = 24^{2}$

Step 5.3

Simplify the right side.

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Step 5.3.1

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

$64 y = 576$

Step 6

Divide each term in $64 y = 576$ by $64$ and simplify.

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Step 6.1

Divide each term in $64 y = 576$ by $64$ .

$\frac{64 y}{64} = \frac{576}{64}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $64$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{64 y}{64} = \frac{576}{64}$

Step 6.2.1.2

Divide $y$ by $1$ .

$y = \frac{576}{64}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Divide $576$ by $64$ .

$y = 9$