Algebra Examples

$\frac{\sqrt{y} + \sqrt{z}}{\sqrt{y} - \sqrt{z}}$

Step 1

Multiply $\frac{\sqrt{y} + \sqrt{z}}{\sqrt{y} - \sqrt{z}}$ by $\frac{\sqrt{y} + \sqrt{z}}{\sqrt{y} + \sqrt{z}}$ .

$\frac{\sqrt{y} + \sqrt{z}}{\sqrt{y} - \sqrt{z}} \cdot \frac{\sqrt{y} + \sqrt{z}}{\sqrt{y} + \sqrt{z}}$

Step 2

Multiply $\frac{\sqrt{y} + \sqrt{z}}{\sqrt{y} - \sqrt{z}}$ by $\frac{\sqrt{y} + \sqrt{z}}{\sqrt{y} + \sqrt{z}}$ .

$\frac{(\sqrt{y} + \sqrt{z}) (\sqrt{y} + \sqrt{z})}{(\sqrt{y} - \sqrt{z}) (\sqrt{y} + \sqrt{z})}$

Step 3

Expand the denominator using the FOIL method.

$\frac{(\sqrt{y} + \sqrt{z}) (\sqrt{y} + \sqrt{z})}{{\sqrt{y}}^{2} + \sqrt{y z} - \sqrt{y z} - {\sqrt{z}}^{2}}$

Step 4

Simplify.

Tap for more steps...

Step 4.1

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

Tap for more steps...

Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.1.3

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

$\frac{(\sqrt{y} + \sqrt{z}) (\sqrt{y} + \sqrt{z})}{y^{\frac{2}{2}} - {\sqrt{z}}^{2}}$

Step 4.1.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.1.4.1

Cancel the common factor.

$\frac{(\sqrt{y} + \sqrt{z}) (\sqrt{y} + \sqrt{z})}{y^{\frac{2}{2}} - {\sqrt{z}}^{2}}$

Step 4.1.4.2

Rewrite the expression.

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

Step 4.1.5

Simplify.

$\frac{(\sqrt{y} + \sqrt{z}) (\sqrt{y} + \sqrt{z})}{y - {\sqrt{z}}^{2}}$

Step 4.2

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

Tap for more steps...

Step 4.2.1

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

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

Step 4.2.2

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

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

Step 4.2.3

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

$\frac{(\sqrt{y} + \sqrt{z}) (\sqrt{y} + \sqrt{z})}{y - z^{\frac{2}{2}}}$

Step 4.2.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.2.4.1

Cancel the common factor.

$\frac{(\sqrt{y} + \sqrt{z}) (\sqrt{y} + \sqrt{z})}{y - z^{\frac{2}{2}}}$

Step 4.2.4.2

Rewrite the expression.

$\frac{(\sqrt{y} + \sqrt{z}) (\sqrt{y} + \sqrt{z})}{y - z^{1}}$

Step 4.2.5

Simplify.

$\frac{(\sqrt{y} + \sqrt{z}) (\sqrt{y} + \sqrt{z})}{y - z}$

Step 5

Simplify the numerator.

Tap for more steps...

Step 5.1

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

$\frac{{(\sqrt{y} + \sqrt{z})}^{1} (\sqrt{y} + \sqrt{z})}{y - z}$

Step 5.2

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

$\frac{{(\sqrt{y} + \sqrt{z})}^{1} {(\sqrt{y} + \sqrt{z})}^{1}}{y - z}$

Step 5.3

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

$\frac{{(\sqrt{y} + \sqrt{z})}^{1 + 1}}{y - z}$

Step 5.4

Add $1$ and $1$ .

$\frac{{(\sqrt{y} + \sqrt{z})}^{2}}{y - z}$

Step 6

Rewrite ${(\sqrt{y} + \sqrt{z})}^{2}$ as $(\sqrt{y} + \sqrt{z}) (\sqrt{y} + \sqrt{z})$ .

$\frac{(\sqrt{y} + \sqrt{z}) (\sqrt{y} + \sqrt{z})}{y - z}$

Step 7

Expand $(\sqrt{y} + \sqrt{z}) (\sqrt{y} + \sqrt{z})$ using the FOIL Method.

Tap for more steps...

Step 7.1

Apply the distributive property.

$\frac{\sqrt{y} (\sqrt{y} + \sqrt{z}) + \sqrt{z} (\sqrt{y} + \sqrt{z})}{y - z}$

Step 7.2

Apply the distributive property.

$\frac{\sqrt{y} \sqrt{y} + \sqrt{y} \sqrt{z} + \sqrt{z} (\sqrt{y} + \sqrt{z})}{y - z}$

Step 7.3

Apply the distributive property.

$\frac{\sqrt{y} \sqrt{y} + \sqrt{y} \sqrt{z} + \sqrt{z} \sqrt{y} + \sqrt{z} \sqrt{z}}{y - z}$

Step 8

Simplify and combine like terms.

Tap for more steps...

Step 8.1

Simplify each term.

Tap for more steps...

Step 8.1.1

Multiply $\sqrt{y} \sqrt{y}$ .

Tap for more steps...

Step 8.1.1.1

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

$\frac{{\sqrt{y}}^{1} \sqrt{y} + \sqrt{y} \sqrt{z} + \sqrt{z} \sqrt{y} + \sqrt{z} \sqrt{z}}{y - z}$

Step 8.1.1.2

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

$\frac{{\sqrt{y}}^{1} {\sqrt{y}}^{1} + \sqrt{y} \sqrt{z} + \sqrt{z} \sqrt{y} + \sqrt{z} \sqrt{z}}{y - z}$

Step 8.1.1.3

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

$\frac{{\sqrt{y}}^{1 + 1} + \sqrt{y} \sqrt{z} + \sqrt{z} \sqrt{y} + \sqrt{z} \sqrt{z}}{y - z}$

Step 8.1.1.4

Add $1$ and $1$ .

$\frac{{\sqrt{y}}^{2} + \sqrt{y} \sqrt{z} + \sqrt{z} \sqrt{y} + \sqrt{z} \sqrt{z}}{y - z}$

Step 8.1.2

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

Tap for more steps...

Step 8.1.2.1

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

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

Step 8.1.2.2

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

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

Step 8.1.2.3

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

$\frac{y^{\frac{2}{2}} + \sqrt{y} \sqrt{z} + \sqrt{z} \sqrt{y} + \sqrt{z} \sqrt{z}}{y - z}$

Step 8.1.2.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 8.1.2.4.1

Cancel the common factor.

$\frac{y^{\frac{2}{2}} + \sqrt{y} \sqrt{z} + \sqrt{z} \sqrt{y} + \sqrt{z} \sqrt{z}}{y - z}$

Step 8.1.2.4.2

Rewrite the expression.

$\frac{y^{1} + \sqrt{y} \sqrt{z} + \sqrt{z} \sqrt{y} + \sqrt{z} \sqrt{z}}{y - z}$

Step 8.1.2.5

Simplify.

$\frac{y + \sqrt{y} \sqrt{z} + \sqrt{z} \sqrt{y} + \sqrt{z} \sqrt{z}}{y - z}$

Step 8.1.3

Combine using the product rule for radicals.

$\frac{y + \sqrt{y z} + \sqrt{z} \sqrt{y} + \sqrt{z} \sqrt{z}}{y - z}$

Step 8.1.4

Combine using the product rule for radicals.

$\frac{y + \sqrt{y z} + \sqrt{z y} + \sqrt{z} \sqrt{z}}{y - z}$

Step 8.1.5

Multiply $\sqrt{z} \sqrt{z}$ .

Tap for more steps...

Step 8.1.5.1

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

$\frac{y + \sqrt{y z} + \sqrt{z y} + {\sqrt{z}}^{1} \sqrt{z}}{y - z}$

Step 8.1.5.2

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

$\frac{y + \sqrt{y z} + \sqrt{z y} + {\sqrt{z}}^{1} {\sqrt{z}}^{1}}{y - z}$

Step 8.1.5.3

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

$\frac{y + \sqrt{y z} + \sqrt{z y} + {\sqrt{z}}^{1 + 1}}{y - z}$

Step 8.1.5.4

Add $1$ and $1$ .

$\frac{y + \sqrt{y z} + \sqrt{z y} + {\sqrt{z}}^{2}}{y - z}$

Step 8.1.6

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

Tap for more steps...

Step 8.1.6.1

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

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

Step 8.1.6.2

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

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

Step 8.1.6.3

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

$\frac{y + \sqrt{y z} + \sqrt{z y} + z^{\frac{2}{2}}}{y - z}$

Step 8.1.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 8.1.6.4.1

Cancel the common factor.

$\frac{y + \sqrt{y z} + \sqrt{z y} + z^{\frac{2}{2}}}{y - z}$

Step 8.1.6.4.2

Rewrite the expression.

$\frac{y + \sqrt{y z} + \sqrt{z y} + z^{1}}{y - z}$

Step 8.1.6.5

Simplify.

$\frac{y + \sqrt{y z} + \sqrt{z y} + z}{y - z}$

Step 8.2

Add $\sqrt{y z}$ and $\sqrt{z y}$ .

Tap for more steps...

Step 8.2.1

Reorder $z$ and $y$ .

$\frac{y + \sqrt{y z} + \sqrt{y z} + z}{y - z}$

Step 8.2.2

Add $\sqrt{y z}$ and $\sqrt{y z}$ .

$\frac{y + 2 \sqrt{y z} + z}{y - z}$