Pre-Algebra Examples

$\sqrt{5 y + 6} = \sqrt{4 y + 11}$

Step 1

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

${\sqrt{5 y + 6}}^{2} = {\sqrt{4 y + 11}}^{2}$

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 $\sqrt{5 y + 6}$ as ${(5 y + 6)}^{\frac{1}{2}}$ .

${({(5 y + 6)}^{\frac{1}{2}})}^{2} = {\sqrt{4 y + 11}}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(5 y + 6)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(5 y + 6)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(5 y + 6)}^{\frac{1}{2} \cdot 2} = {\sqrt{4 y + 11}}^{2}$

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.

${(5 y + 6)}^{\frac{1}{2} \cdot 2} = {\sqrt{4 y + 11}}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(5 y + 6)}^{1} = {\sqrt{4 y + 11}}^{2}$

Step 2.2.1.2

Simplify.

$5 y + 6 = {\sqrt{4 y + 11}}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Rewrite ${\sqrt{4 y + 11}}^{2}$ as $4 y + 11$ .

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Step 2.3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 y + 11}$ as ${(4 y + 11)}^{\frac{1}{2}}$ .

$5 y + 6 = {({(4 y + 11)}^{\frac{1}{2}})}^{2}$

Step 2.3.1.2

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

$5 y + 6 = {(4 y + 11)}^{\frac{1}{2} \cdot 2}$

Step 2.3.1.3

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

$5 y + 6 = {(4 y + 11)}^{\frac{2}{2}}$

Step 2.3.1.4

Cancel the common factor of $2$ .

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Step 2.3.1.4.1

Cancel the common factor.

$5 y + 6 = {(4 y + 11)}^{\frac{2}{2}}$

Step 2.3.1.4.2

Rewrite the expression.

$5 y + 6 = {(4 y + 11)}^{1}$

Step 2.3.1.5

Simplify.

$5 y + 6 = 4 y + 11$

Step 3

Solve for $y$ .

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Step 3.1

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

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Step 3.1.1

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

$5 y + 6 - 4 y = 11$

Step 3.1.2

Subtract $4 y$ from $5 y$ .

$y + 6 = 11$

Step 3.2

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

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Step 3.2.1

Subtract $6$ from both sides of the equation.

$y = 11 - 6$

Step 3.2.2

Subtract $6$ from $11$ .

$y = 5$