Calculus Examples

$0 = \sqrt{x - 1} + \sqrt{y}$

Step 1

Rewrite the equation as $\sqrt{x - 1} + \sqrt{y} = 0$ .

$\sqrt{x - 1} + \sqrt{y} = 0$

Step 2

Subtract $\sqrt{y}$ from both sides of the equation.

$\sqrt{x - 1} = - \sqrt{y}$

Step 3

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

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

Step 4

Simplify each side of the equation.

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Step 4.1

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

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

Step 4.2

Simplify the left side.

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Step 4.2.1

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

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Step 4.2.1.1

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

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Step 4.2.1.1.1

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

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

Step 4.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.1.1.2.1

Cancel the common factor.

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

Step 4.2.1.1.2.2

Rewrite the expression.

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

Step 4.2.1.2

Simplify.

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

Step 4.3

Simplify the right side.

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Step 4.3.1

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

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Step 4.3.1.1

Simplify the expression.

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Step 4.3.1.1.1

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

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

Step 4.3.1.1.2

Raise $- 1$ to the power of $2$ .

$x - 1 = 1 {\sqrt{y}}^{2}$

Step 4.3.1.1.3

Multiply ${\sqrt{y}}^{2}$ by $1$ .

$x - 1 = {\sqrt{y}}^{2}$

Step 4.3.1.2

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

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Step 4.3.1.2.1

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

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

Step 4.3.1.2.2

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

$x - 1 = y^{\frac{1}{2} \cdot 2}$

Step 4.3.1.2.3

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

$x - 1 = y^{\frac{2}{2}}$

Step 4.3.1.2.4

Cancel the common factor of $2$ .

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Step 4.3.1.2.4.1

Cancel the common factor.

$x - 1 = y^{\frac{2}{2}}$

Step 4.3.1.2.4.2

Rewrite the expression.

$x - 1 = y^{1}$

Step 4.3.1.2.5

Simplify.

$x - 1 = y$

Step 5

Add $1$ to both sides of the equation.

$x = y + 1$