Algebra Examples

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

Step 1

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

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

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 $\sqrt{y^{2} + 4}$ as ${(y^{2} + 4)}^{\frac{1}{2}}$ .

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

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

Multiply the exponents in ${({(y^{2} + 4)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

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

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.

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

Step 2.2.1.1.2.2

Rewrite the expression.

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

Step 2.2.1.2

Simplify.

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

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${\sqrt{4 y}}^{2}$ .

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Step 2.3.1.1

Rewrite $4$ as $2^{2}$ .

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

Step 2.3.1.2

Pull terms out from under the radical.

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

Step 2.3.1.3

Simplify the expression.

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Step 2.3.1.3.1

Apply the product rule to $2 \sqrt{y}$ .

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

Step 2.3.1.3.2

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

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

Step 2.3.1.4

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

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Step 2.3.1.4.1

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

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

Step 2.3.1.4.2

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

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

Step 2.3.1.4.3

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

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

Step 2.3.1.4.4

Cancel the common factor of $2$ .

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Step 2.3.1.4.4.1

Cancel the common factor.

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

Step 2.3.1.4.4.2

Rewrite the expression.

$y^{2} + 4 = 4 y^{1}$

Step 2.3.1.4.5

Simplify.

$y^{2} + 4 = 4 y$

Step 3

Solve for $y$ .

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Step 3.1

Subtract $4 y$ from both sides of the equation.

$y^{2} + 4 - 4 y = 0$

Step 3.2

Factor using the perfect square rule.

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Step 3.2.1

Rearrange terms.

$y^{2} - 4 y + 4 = 0$

Step 3.2.2

Rewrite $4$ as $2^{2}$ .

$y^{2} - 4 y + 2^{2} = 0$

Step 3.2.3

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 y = 2 \cdot y \cdot 2$

Step 3.2.4

Rewrite the polynomial.

$y^{2} - 2 \cdot y \cdot 2 + 2^{2} = 0$

Step 3.2.5

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = y$ and $b = 2$ .

${(y - 2)}^{2} = 0$

Step 3.3

Set the $y - 2$ equal to $0$ .

$y - 2 = 0$

Step 3.4

Add $2$ to both sides of the equation.

$y = 2$