Calculus Examples

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

Step 1

Move $x^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{\sqrt{7}}{4 \sqrt{2 + \sqrt{7 x}} x^{\frac{1}{2}}}$

Step 2

Multiply $\frac{\sqrt{7}}{4 \sqrt{2 + \sqrt{7 x}} x^{\frac{1}{2}}}$ by $\frac{\sqrt{2 + \sqrt{7 x}}}{\sqrt{2 + \sqrt{7 x}}}$ .

$\frac{\sqrt{7}}{4 \sqrt{2 + \sqrt{7 x}} x^{\frac{1}{2}}} \cdot \frac{\sqrt{2 + \sqrt{7 x}}}{\sqrt{2 + \sqrt{7 x}}}$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{\sqrt{7}}{4 \sqrt{2 + \sqrt{7 x}} x^{\frac{1}{2}}}$ by $\frac{\sqrt{2 + \sqrt{7 x}}}{\sqrt{2 + \sqrt{7 x}}}$ .

$\frac{\sqrt{7} \sqrt{2 + \sqrt{7 x}}}{4 \sqrt{2 + \sqrt{7 x}} x^{\frac{1}{2}} \sqrt{2 + \sqrt{7 x}}}$

Step 3.2

Move $\sqrt{2 + \sqrt{7 x}}$ .

$\frac{\sqrt{7} \sqrt{2 + \sqrt{7 x}}}{4 x^{\frac{1}{2}} (\sqrt{2 + \sqrt{7 x}} \sqrt{2 + \sqrt{7 x}})}$

Step 3.3

Raise $\sqrt{2 + \sqrt{7 x}}$ to the power of $1$ .

$\frac{\sqrt{7} \sqrt{2 + \sqrt{7 x}}}{4 x^{\frac{1}{2}} ({\sqrt{2 + \sqrt{7 x}}}^{1} \sqrt{2 + \sqrt{7 x}})}$

Step 3.4

Raise $\sqrt{2 + \sqrt{7 x}}$ to the power of $1$ .

$\frac{\sqrt{7} \sqrt{2 + \sqrt{7 x}}}{4 x^{\frac{1}{2}} ({\sqrt{2 + \sqrt{7 x}}}^{1} {\sqrt{2 + \sqrt{7 x}}}^{1})}$

Step 3.5

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

$\frac{\sqrt{7} \sqrt{2 + \sqrt{7 x}}}{4 x^{\frac{1}{2}} {\sqrt{2 + \sqrt{7 x}}}^{1 + 1}}$

Step 3.6

Add $1$ and $1$ .

$\frac{\sqrt{7} \sqrt{2 + \sqrt{7 x}}}{4 x^{\frac{1}{2}} {\sqrt{2 + \sqrt{7 x}}}^{2}}$

Step 3.7

Rewrite ${\sqrt{2 + \sqrt{7 x}}}^{2}$ as $2 + \sqrt{7 x}$ .

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Step 3.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 + \sqrt{7 x}}$ as ${(2 + \sqrt{7 x})}^{\frac{1}{2}}$ .

$\frac{\sqrt{7} \sqrt{2 + \sqrt{7 x}}}{4 x^{\frac{1}{2}} {({(2 + \sqrt{7 x})}^{\frac{1}{2}})}^{2}}$

Step 3.7.2

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

$\frac{\sqrt{7} \sqrt{2 + \sqrt{7 x}}}{4 x^{\frac{1}{2}} {(2 + \sqrt{7 x})}^{\frac{1}{2} \cdot 2}}$

Step 3.7.3

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

$\frac{\sqrt{7} \sqrt{2 + \sqrt{7 x}}}{4 x^{\frac{1}{2}} {(2 + \sqrt{7 x})}^{\frac{2}{2}}}$

Step 3.7.4

Cancel the common factor of $2$ .

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Step 3.7.4.1

Cancel the common factor.

$\frac{\sqrt{7} \sqrt{2 + \sqrt{7 x}}}{4 x^{\frac{1}{2}} {(2 + \sqrt{7 x})}^{\frac{2}{2}}}$

Step 3.7.4.2

Rewrite the expression.

$\frac{\sqrt{7} \sqrt{2 + \sqrt{7 x}}}{4 x^{\frac{1}{2}} {(2 + \sqrt{7 x})}^{1}}$

Step 3.7.5

Simplify.

$\frac{\sqrt{7} \sqrt{2 + \sqrt{7 x}}}{4 x^{\frac{1}{2}} (2 + \sqrt{7 x})}$

Step 4

Combine using the product rule for radicals.

$\frac{\sqrt{7 (2 + \sqrt{7 x})}}{4 x^{\frac{1}{2}} (2 + \sqrt{7 x})}$

Step 5

Multiply $\frac{\sqrt{7 (2 + \sqrt{7 x})}}{4 x^{\frac{1}{2}} (2 + \sqrt{7 x})}$ by $\frac{2 - \sqrt{7 x}}{2 - \sqrt{7 x}}$ .

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

Step 6

Simplify terms.

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Step 6.1

Multiply $\frac{\sqrt{7 (2 + \sqrt{7 x})}}{4 x^{\frac{1}{2}} (2 + \sqrt{7 x})}$ by $\frac{2 - \sqrt{7 x}}{2 - \sqrt{7 x}}$ .

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

Step 6.2

Reorder.

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Step 6.2.1

Move $2 + \sqrt{7 x}$ .

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

Step 6.2.2

Expand the denominator using the FOIL method.

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

Step 6.2.3

Simplify.

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Step 6.2.3.1

Rewrite ${\sqrt{7 x}}^{2}$ as $7 x$ .

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Step 6.2.3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7 x}$ as ${(7 x)}^{\frac{1}{2}}$ .

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

Step 6.2.3.1.2

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

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

Step 6.2.3.1.3

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

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

Step 6.2.3.1.4

Cancel the common factor of $2$ .

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Step 6.2.3.1.4.1

Cancel the common factor.

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

Step 6.2.3.1.4.2

Rewrite the expression.

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

Step 6.2.3.1.5

Simplify.

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

Step 6.2.3.2

Multiply $7$ by $- 1$ .

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

Step 6.3

Apply the distributive property.

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

Step 6.4

Reorder.

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Step 6.4.1

Move $2$ to the left of $\sqrt{7 (2 + \sqrt{7 x})}$ .

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

Step 6.4.2

Rewrite using the commutative property of multiplication.

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

Step 7

Simplify each term.

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Step 7.1

Multiply $- \sqrt{7 (2 + \sqrt{7 x})} \sqrt{7 x}$ .

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Step 7.1.1

Combine using the product rule for radicals.

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

Step 7.1.2

Multiply $7$ by $7$ .

$\frac{2 \sqrt{7 (2 + \sqrt{7 x})} - \sqrt{49 x (2 + \sqrt{7 x})}}{4 x^{\frac{1}{2}} (4 - 7 x)}$

Step 7.2

Rewrite $49 x (2 + \sqrt{7 x})$ as $7^{2} (x (2 + \sqrt{7 x}))$ .

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Step 7.2.1

Rewrite $49$ as $7^{2}$ .

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

Step 7.2.2

Add parentheses.

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

Step 7.3

Pull terms out from under the radical.

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

Step 7.4

Multiply $7$ by $- 1$ .

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