Basic Math Examples

$\sqrt{y + 7} = \sqrt{y} - 7$

Step 1

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

${\sqrt{y + 7}}^{2} = {(\sqrt{y} - 7)}^{2}$

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 $\sqrt{y + 7}$ as ${(y + 7)}^{\frac{1}{2}}$ .

${({(y + 7)}^{\frac{1}{2}})}^{2} = {(\sqrt{y} - 7)}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(y + 7)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(y + 7)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(y + 7)}^{\frac{1}{2} \cdot 2} = {(\sqrt{y} - 7)}^{2}$

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.

${(y + 7)}^{\frac{1}{2} \cdot 2} = {(\sqrt{y} - 7)}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(y + 7)}^{1} = {(\sqrt{y} - 7)}^{2}$

Step 2.2.1.2

Simplify.

$y + 7 = {(\sqrt{y} - 7)}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${(\sqrt{y} - 7)}^{2}$ .

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Step 2.3.1.1

Rewrite ${(\sqrt{y} - 7)}^{2}$ as $(\sqrt{y} - 7) (\sqrt{y} - 7)$ .

$y + 7 = (\sqrt{y} - 7) (\sqrt{y} - 7)$

Step 2.3.1.2

Expand $(\sqrt{y} - 7) (\sqrt{y} - 7)$ using the FOIL Method.

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Step 2.3.1.2.1

Apply the distributive property.

$y + 7 = \sqrt{y} (\sqrt{y} - 7) - 7 (\sqrt{y} - 7)$

Step 2.3.1.2.2

Apply the distributive property.

$y + 7 = \sqrt{y} \sqrt{y} + \sqrt{y} \cdot - 7 - 7 (\sqrt{y} - 7)$

Step 2.3.1.2.3

Apply the distributive property.

$y + 7 = \sqrt{y} \sqrt{y} + \sqrt{y} \cdot - 7 - 7 \sqrt{y} - 7 \cdot - 7$

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 $1$ .

$y + 7 = {\sqrt{y}}^{1} \sqrt{y} + \sqrt{y} \cdot - 7 - 7 \sqrt{y} - 7 \cdot - 7$

Step 2.3.1.3.1.1.2

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

$y + 7 = {\sqrt{y}}^{1} {\sqrt{y}}^{1} + \sqrt{y} \cdot - 7 - 7 \sqrt{y} - 7 \cdot - 7$

Step 2.3.1.3.1.1.3

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

$y + 7 = {\sqrt{y}}^{1 + 1} + \sqrt{y} \cdot - 7 - 7 \sqrt{y} - 7 \cdot - 7$

Step 2.3.1.3.1.1.4

Add $1$ and $1$ .

$y + 7 = {\sqrt{y}}^{2} + \sqrt{y} \cdot - 7 - 7 \sqrt{y} - 7 \cdot - 7$

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

$y + 7 = {(y^{\frac{1}{2}})}^{2} + \sqrt{y} \cdot - 7 - 7 \sqrt{y} - 7 \cdot - 7$

Step 2.3.1.3.1.2.2

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

$y + 7 = y^{\frac{1}{2} \cdot 2} + \sqrt{y} \cdot - 7 - 7 \sqrt{y} - 7 \cdot - 7$

Step 2.3.1.3.1.2.3

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

$y + 7 = y^{\frac{2}{2}} + \sqrt{y} \cdot - 7 - 7 \sqrt{y} - 7 \cdot - 7$

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.

$y + 7 = y^{\frac{2}{2}} + \sqrt{y} \cdot - 7 - 7 \sqrt{y} - 7 \cdot - 7$

Step 2.3.1.3.1.2.4.2

Rewrite the expression.

$y + 7 = y^{1} + \sqrt{y} \cdot - 7 - 7 \sqrt{y} - 7 \cdot - 7$

Step 2.3.1.3.1.2.5

Simplify.

$y + 7 = y + \sqrt{y} \cdot - 7 - 7 \sqrt{y} - 7 \cdot - 7$

Step 2.3.1.3.1.3

Move $- 7$ to the left of $\sqrt{y}$ .

$y + 7 = y - 7 \cdot \sqrt{y} - 7 \sqrt{y} - 7 \cdot - 7$

Step 2.3.1.3.1.4

Multiply $- 7$ by $- 7$ .

$y + 7 = y - 7 \sqrt{y} - 7 \sqrt{y} + 49$

Step 2.3.1.3.2

Subtract $7 \sqrt{y}$ from $- 7 \sqrt{y}$ .

$y + 7 = y - 14 \sqrt{y} + 49$

Step 3

Solve for $- 14 \sqrt{y}$ .

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Step 3.1

Rewrite the equation as $y - 14 \sqrt{y} + 49 = y + 7$ .

$y - 14 \sqrt{y} + 49 = y + 7$

Step 3.2

Move all terms not containing $- 14 \sqrt{y}$ to the right side of the equation.

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Step 3.2.1

Subtract $y$ from both sides of the equation.

$- 14 \sqrt{y} + 49 = y + 7 - y$

Step 3.2.2

Subtract $49$ from both sides of the equation.

$- 14 \sqrt{y} = y + 7 - y - 49$

Step 3.2.3

Combine the opposite terms in $y + 7 - y - 49$ .

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Step 3.2.3.1

Subtract $y$ from $y$ .

$- 14 \sqrt{y} = 0 + 7 - 49$

Step 3.2.3.2

Add $0$ and $7$ .

$- 14 \sqrt{y} = 7 - 49$

Step 3.2.4

Subtract $49$ from $7$ .

$- 14 \sqrt{y} = - 42$

Step 4

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

${(- 14 \sqrt{y})}^{2} = {(- 42)}^{2}$

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

${(- 14 y^{\frac{1}{2}})}^{2} = {(- 42)}^{2}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Simplify ${(- 14 y^{\frac{1}{2}})}^{2}$ .

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Step 5.2.1.1

Apply the product rule to $- 14 y^{\frac{1}{2}}$ .

${(- 14)}^{2} {(y^{\frac{1}{2}})}^{2} = {(- 42)}^{2}$

Step 5.2.1.2

Raise $- 14$ to the power of $2$ .

$196 {(y^{\frac{1}{2}})}^{2} = {(- 42)}^{2}$

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, ${(a^{m})}^{n} = a^{m n}$ .

$196 y^{\frac{1}{2} \cdot 2} = {(- 42)}^{2}$

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.

$196 y^{\frac{1}{2} \cdot 2} = {(- 42)}^{2}$

Step 5.2.1.3.2.2

Rewrite the expression.

$196 y^{1} = {(- 42)}^{2}$

Step 5.2.1.4

Simplify.

$196 y = {(- 42)}^{2}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Raise $- 42$ to the power of $2$ .

$196 y = 1764$

Step 6

Divide each term in $196 y = 1764$ by $196$ and simplify.

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Step 6.1

Divide each term in $196 y = 1764$ by $196$ .

$\frac{196 y}{196} = \frac{1764}{196}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $196$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{196 y}{196} = \frac{1764}{196}$

Step 6.2.1.2

Divide $y$ by $1$ .

$y = \frac{1764}{196}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Divide $1764$ by $196$ .

$y = 9$

Step 7

Exclude the solutions that do not make $\sqrt{y + 7} = \sqrt{y} - 7$ true.

$No solution$