Pre-Algebra Examples

$\sqrt{y + 28} - \sqrt{y} = 2$

Step 1

Add $\sqrt{y}$ to both sides of the equation.

$\sqrt{y + 28} = 2 + \sqrt{y}$

Step 2

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

${\sqrt{y + 28}}^{2} = {(2 + \sqrt{y})}^{2}$

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 $\sqrt{y + 28}$ as ${(y + 28)}^{\frac{1}{2}}$ .

${({(y + 28)}^{\frac{1}{2}})}^{2} = {(2 + \sqrt{y})}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(y + 28)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(y + 28)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(y + 28)}^{\frac{1}{2} \cdot 2} = {(2 + \sqrt{y})}^{2}$

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.

${(y + 28)}^{\frac{1}{2} \cdot 2} = {(2 + \sqrt{y})}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(y + 28)}^{1} = {(2 + \sqrt{y})}^{2}$

Step 3.2.1.2

Simplify.

$y + 28 = {(2 + \sqrt{y})}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

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

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Step 3.3.1.1

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

$y + 28 = (2 + \sqrt{y}) (2 + \sqrt{y})$

Step 3.3.1.2

Expand $(2 + \sqrt{y}) (2 + \sqrt{y})$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

$y + 28 = 2 (2 + \sqrt{y}) + \sqrt{y} (2 + \sqrt{y})$

Step 3.3.1.2.2

Apply the distributive property.

$y + 28 = 2 \cdot 2 + 2 \sqrt{y} + \sqrt{y} (2 + \sqrt{y})$

Step 3.3.1.2.3

Apply the distributive property.

$y + 28 = 2 \cdot 2 + 2 \sqrt{y} + \sqrt{y} \cdot 2 + \sqrt{y} \sqrt{y}$

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 $2$ by $2$ .

$y + 28 = 4 + 2 \sqrt{y} + \sqrt{y} \cdot 2 + \sqrt{y} \sqrt{y}$

Step 3.3.1.3.1.2

Move $2$ to the left of $\sqrt{y}$ .

$y + 28 = 4 + 2 \sqrt{y} + 2 \cdot \sqrt{y} + \sqrt{y} \sqrt{y}$

Step 3.3.1.3.1.3

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

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Step 3.3.1.3.1.3.1

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

$y + 28 = 4 + 2 \sqrt{y} + 2 \sqrt{y} + {\sqrt{y}}^{1} \sqrt{y}$

Step 3.3.1.3.1.3.2

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

$y + 28 = 4 + 2 \sqrt{y} + 2 \sqrt{y} + {\sqrt{y}}^{1} {\sqrt{y}}^{1}$

Step 3.3.1.3.1.3.3

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

$y + 28 = 4 + 2 \sqrt{y} + 2 \sqrt{y} + {\sqrt{y}}^{1 + 1}$

Step 3.3.1.3.1.3.4

Add $1$ and $1$ .

$y + 28 = 4 + 2 \sqrt{y} + 2 \sqrt{y} + {\sqrt{y}}^{2}$

Step 3.3.1.3.1.4

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

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Step 3.3.1.3.1.4.1

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

$y + 28 = 4 + 2 \sqrt{y} + 2 \sqrt{y} + {(y^{\frac{1}{2}})}^{2}$

Step 3.3.1.3.1.4.2

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

$y + 28 = 4 + 2 \sqrt{y} + 2 \sqrt{y} + y^{\frac{1}{2} \cdot 2}$

Step 3.3.1.3.1.4.3

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

$y + 28 = 4 + 2 \sqrt{y} + 2 \sqrt{y} + y^{\frac{2}{2}}$

Step 3.3.1.3.1.4.4

Cancel the common factor of $2$ .

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Step 3.3.1.3.1.4.4.1

Cancel the common factor.

$y + 28 = 4 + 2 \sqrt{y} + 2 \sqrt{y} + y^{\frac{2}{2}}$

Step 3.3.1.3.1.4.4.2

Rewrite the expression.

$y + 28 = 4 + 2 \sqrt{y} + 2 \sqrt{y} + y^{1}$

Step 3.3.1.3.1.4.5

Simplify.

$y + 28 = 4 + 2 \sqrt{y} + 2 \sqrt{y} + y$

Step 3.3.1.3.2

Add $2 \sqrt{y}$ and $2 \sqrt{y}$ .

$y + 28 = 4 + 4 \sqrt{y} + y$

Step 4

Solve for $4 \sqrt{y}$ .

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Step 4.1

Rewrite the equation as $4 + 4 \sqrt{y} + y = y + 28$ .

$4 + 4 \sqrt{y} + y = y + 28$

Step 4.2

Move all terms not containing $4 \sqrt{y}$ to the right side of the equation.

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Step 4.2.1

Subtract $4$ from both sides of the equation.

$4 \sqrt{y} + y = y + 28 - 4$

Step 4.2.2

Subtract $y$ from both sides of the equation.

$4 \sqrt{y} = y + 28 - 4 - y$

Step 4.2.3

Combine the opposite terms in $y + 28 - 4 - y$ .

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Step 4.2.3.1

Subtract $y$ from $y$ .

$4 \sqrt{y} = 0 + 28 - 4$

Step 4.2.3.2

Add $0$ and $28$ .

$4 \sqrt{y} = 28 - 4$

Step 4.2.4

Subtract $4$ from $28$ .

$4 \sqrt{y} = 24$

Step 5

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

${(4 \sqrt{y})}^{2} = 24^{2}$

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

${(4 y^{\frac{1}{2}})}^{2} = 24^{2}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Simplify ${(4 y^{\frac{1}{2}})}^{2}$ .

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Step 6.2.1.1

Apply the product rule to $4 y^{\frac{1}{2}}$ .

$4^{2} {(y^{\frac{1}{2}})}^{2} = 24^{2}$

Step 6.2.1.2

Raise $4$ to the power of $2$ .

$16 {(y^{\frac{1}{2}})}^{2} = 24^{2}$

Step 6.2.1.3

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

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Step 6.2.1.3.1

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

$16 y^{\frac{1}{2} \cdot 2} = 24^{2}$

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.

$16 y^{\frac{1}{2} \cdot 2} = 24^{2}$

Step 6.2.1.3.2.2

Rewrite the expression.

$16 y^{1} = 24^{2}$

Step 6.2.1.4

Simplify.

$16 y = 24^{2}$

Step 6.3

Simplify the right side.

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Step 6.3.1

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

$16 y = 576$

Step 7

Divide each term in $16 y = 576$ by $16$ and simplify.

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Step 7.1

Divide each term in $16 y = 576$ by $16$ .

$\frac{16 y}{16} = \frac{576}{16}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $16$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{16 y}{16} = \frac{576}{16}$

Step 7.2.1.2

Divide $y$ by $1$ .

$y = \frac{576}{16}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Divide $576$ by $16$ .

$y = 36$