Basic Math Examples

$\sqrt{2 y + y} = \sqrt{2 y - 8}$

Step 1

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

${\sqrt{2 y + y}}^{2} = {\sqrt{2 y - 8}}^{2}$

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

${({(2 y + y)}^{\frac{1}{2}})}^{2} = {\sqrt{2 y - 8}}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

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

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Step 2.2.1.1.1

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

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

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.

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

Step 2.2.1.1.2.2

Rewrite the expression.

${(2 y + y)}^{1} = {\sqrt{2 y - 8}}^{2}$

Step 2.2.1.2

Add $2 y$ and $y$ .

${(3 y)}^{1} = {\sqrt{2 y - 8}}^{2}$

Step 2.2.1.3

Simplify.

$3 y = {\sqrt{2 y - 8}}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${\sqrt{2 y - 8}}^{2}$ .

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Step 2.3.1.1

Factor $2$ out of $2 y - 8$ .

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Step 2.3.1.1.1

Factor $2$ out of $2 y$ .

$3 y = {\sqrt{2 (y) - 8}}^{2}$

Step 2.3.1.1.2

Factor $2$ out of $- 8$ .

$3 y = {\sqrt{2 y + 2 \cdot - 4}}^{2}$

Step 2.3.1.1.3

Factor $2$ out of $2 y + 2 \cdot - 4$ .

$3 y = {\sqrt{2 (y - 4)}}^{2}$

Step 2.3.1.2

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

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Step 2.3.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 (y - 4)}$ as ${(2 (y - 4))}^{\frac{1}{2}}$ .

$3 y = {({(2 (y - 4))}^{\frac{1}{2}})}^{2}$

Step 2.3.1.2.2

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

$3 y = {(2 (y - 4))}^{\frac{1}{2} \cdot 2}$

Step 2.3.1.2.3

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

$3 y = {(2 (y - 4))}^{\frac{2}{2}}$

Step 2.3.1.2.4

Cancel the common factor of $2$ .

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Step 2.3.1.2.4.1

Cancel the common factor.

$3 y = {(2 (y - 4))}^{\frac{2}{2}}$

Step 2.3.1.2.4.2

Rewrite the expression.

$3 y = {(2 (y - 4))}^{1}$

Step 2.3.1.2.5

Simplify.

$3 y = 2 (y - 4)$

Step 2.3.1.3

Apply the distributive property.

$3 y = 2 y + 2 \cdot - 4$

Step 2.3.1.4

Multiply $2$ by $- 4$ .

$3 y = 2 y - 8$

Step 3

Move all terms containing $y$ to the left side of the equation.

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Step 3.1

Subtract $2 y$ from both sides of the equation.

$3 y - 2 y = - 8$

Step 3.2

Subtract $2 y$ from $3 y$ .

$y = - 8$

Step 4

Exclude the solutions that do not make $\sqrt{2 y + y} = \sqrt{2 y - 8}$ true.

$No solution$