Basic Math Examples

$\frac{7 \sqrt{x}}{4 \sqrt{x} + \sqrt{y}}$

Step 1

Multiply $\frac{7 \sqrt{x}}{4 \sqrt{x} + \sqrt{y}}$ by $\frac{4 \sqrt{x} - \sqrt{y}}{4 \sqrt{x} - \sqrt{y}}$ .

$\frac{7 \sqrt{x}}{4 \sqrt{x} + \sqrt{y}} \cdot \frac{4 \sqrt{x} - \sqrt{y}}{4 \sqrt{x} - \sqrt{y}}$

Step 2

Multiply $\frac{7 \sqrt{x}}{4 \sqrt{x} + \sqrt{y}}$ by $\frac{4 \sqrt{x} - \sqrt{y}}{4 \sqrt{x} - \sqrt{y}}$ .

$\frac{7 \sqrt{x} (4 \sqrt{x} - \sqrt{y})}{(4 \sqrt{x} + \sqrt{y}) (4 \sqrt{x} - \sqrt{y})}$

Step 3

Expand the denominator using the FOIL method.

$\frac{7 \sqrt{x} (4 \sqrt{x} - \sqrt{y})}{16 {\sqrt{x}}^{2} - 4 \sqrt{y x} + 4 \sqrt{y x} - {\sqrt{y}}^{2}}$

Step 4

Simplify.

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Step 4.1

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

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Step 4.1.1

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

$\frac{7 \sqrt{x} (4 \sqrt{x} - \sqrt{y})}{16 {(x^{\frac{1}{2}})}^{2} - {\sqrt{y}}^{2}}$

Step 4.1.2

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

$\frac{7 \sqrt{x} (4 \sqrt{x} - \sqrt{y})}{16 x^{\frac{1}{2} \cdot 2} - {\sqrt{y}}^{2}}$

Step 4.1.3

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

$\frac{7 \sqrt{x} (4 \sqrt{x} - \sqrt{y})}{16 x^{\frac{2}{2}} - {\sqrt{y}}^{2}}$

Step 4.1.4

Cancel the common factor of $2$ .

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Step 4.1.4.1

Cancel the common factor.

$\frac{7 \sqrt{x} (4 \sqrt{x} - \sqrt{y})}{16 x^{\frac{2}{2}} - {\sqrt{y}}^{2}}$

Step 4.1.4.2

Rewrite the expression.

$\frac{7 \sqrt{x} (4 \sqrt{x} - \sqrt{y})}{16 x^{1} - {\sqrt{y}}^{2}}$

Step 4.1.5

Simplify.

$\frac{7 \sqrt{x} (4 \sqrt{x} - \sqrt{y})}{16 x - {\sqrt{y}}^{2}}$

Step 4.2

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

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Step 4.2.1

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

$\frac{7 \sqrt{x} (4 \sqrt{x} - \sqrt{y})}{16 x - {(y^{\frac{1}{2}})}^{2}}$

Step 4.2.2

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

$\frac{7 \sqrt{x} (4 \sqrt{x} - \sqrt{y})}{16 x - y^{\frac{1}{2} \cdot 2}}$

Step 4.2.3

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

$\frac{7 \sqrt{x} (4 \sqrt{x} - \sqrt{y})}{16 x - y^{\frac{2}{2}}}$

Step 4.2.4

Cancel the common factor of $2$ .

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Step 4.2.4.1

Cancel the common factor.

$\frac{7 \sqrt{x} (4 \sqrt{x} - \sqrt{y})}{16 x - y^{\frac{2}{2}}}$

Step 4.2.4.2

Rewrite the expression.

$\frac{7 \sqrt{x} (4 \sqrt{x} - \sqrt{y})}{16 x - y^{1}}$

Step 4.2.5

Simplify.

$\frac{7 \sqrt{x} (4 \sqrt{x} - \sqrt{y})}{16 x - y}$

Step 5

Group $4 \sqrt{x} - \sqrt{y}$ and $\sqrt{x}$ together.

$\frac{7 ((4 \sqrt{x} - \sqrt{y}) \sqrt{x})}{16 x - y}$

Step 6

Apply the distributive property.

$\frac{7 (4 \sqrt{x} \sqrt{x} - \sqrt{y} \sqrt{x})}{16 x - y}$

Step 7

Multiply $4 \sqrt{x} \sqrt{x}$ .

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Step 7.1

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

$\frac{7 (4 ({\sqrt{x}}^{1} \sqrt{x}) - \sqrt{y} \sqrt{x})}{16 x - y}$

Step 7.2

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

$\frac{7 (4 ({\sqrt{x}}^{1} {\sqrt{x}}^{1}) - \sqrt{y} \sqrt{x})}{16 x - y}$

Step 7.3

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

$\frac{7 (4 {\sqrt{x}}^{1 + 1} - \sqrt{y} \sqrt{x})}{16 x - y}$

Step 7.4

Add $1$ and $1$ .

$\frac{7 (4 {\sqrt{x}}^{2} - \sqrt{y} \sqrt{x})}{16 x - y}$

Step 8

Simplify terms.

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Step 8.1

Combine using the product rule for radicals.

$\frac{7 (4 {\sqrt{x}}^{2} - \sqrt{x y})}{16 x - y}$

Step 8.2

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

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Step 8.2.1

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

$\frac{7 (4 {(x^{\frac{1}{2}})}^{2} - \sqrt{x y})}{16 x - y}$

Step 8.2.2

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

$\frac{7 (4 x^{\frac{1}{2} \cdot 2} - \sqrt{x y})}{16 x - y}$

Step 8.2.3

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

$\frac{7 (4 x^{\frac{2}{2}} - \sqrt{x y})}{16 x - y}$

Step 8.2.4

Cancel the common factor of $2$ .

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Step 8.2.4.1

Cancel the common factor.

$\frac{7 (4 x^{\frac{2}{2}} - \sqrt{x y})}{16 x - y}$

Step 8.2.4.2

Rewrite the expression.

$\frac{7 (4 x^{1} - \sqrt{x y})}{16 x - y}$

Step 8.2.5

Simplify.

$\frac{7 (4 x - \sqrt{x y})}{16 x - y}$

Step 9

Simplify the numerator.

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Step 9.1

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

$\frac{7 (4 x - {(x y)}^{\frac{1}{2}})}{16 x - y}$

Step 9.2

Apply the product rule to $x y$ .

$\frac{7 (4 x - (x^{\frac{1}{2}} y^{\frac{1}{2}}))}{16 x - y}$

Step 9.3

Factor $x^{\frac{1}{2}}$ out of $4 x - x^{\frac{1}{2}} y^{\frac{1}{2}}$ .

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Step 9.3.1

Factor $x^{\frac{1}{2}}$ out of $4 x$ .

$\frac{7 (x^{\frac{1}{2}} (4 x^{\frac{1}{2}}) - x^{\frac{1}{2}} y^{\frac{1}{2}})}{16 x - y}$

Step 9.3.2

Factor $x^{\frac{1}{2}}$ out of $- x^{\frac{1}{2}} y^{\frac{1}{2}}$ .

$\frac{7 (x^{\frac{1}{2}} (4 x^{\frac{1}{2}}) + x^{\frac{1}{2}} (- y^{\frac{1}{2}}))}{16 x - y}$

Step 9.3.3

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{1}{2}} (4 x^{\frac{1}{2}}) + x^{\frac{1}{2}} (- y^{\frac{1}{2}})$ .

$\frac{7 (x^{\frac{1}{2}} (4 x^{\frac{1}{2}} - y^{\frac{1}{2}}))}{16 x - y}$

$\frac{7 x^{\frac{1}{2}} (4 x^{\frac{1}{2}} - y^{\frac{1}{2}})}{16 x - y}$