Calculus Examples

$\sqrt{\frac{99}{\sqrt[3]{198 p^{2}}}}$

Step 1

Multiply $\frac{99}{\sqrt[3]{198 p^{2}}}$ by $\frac{{\sqrt[3]{198 p^{2}}}^{2}}{{\sqrt[3]{198 p^{2}}}^{2}}$ .

$\sqrt{\frac{99}{\sqrt[3]{198 p^{2}}} \cdot \frac{{\sqrt[3]{198 p^{2}}}^{2}}{{\sqrt[3]{198 p^{2}}}^{2}}}$

Step 2

Simplify terms.

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Step 2.1

Combine and simplify the denominator.

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Step 2.1.1

Multiply $\frac{99}{\sqrt[3]{198 p^{2}}}$ by $\frac{{\sqrt[3]{198 p^{2}}}^{2}}{{\sqrt[3]{198 p^{2}}}^{2}}$ .

$\sqrt{\frac{99 {\sqrt[3]{198 p^{2}}}^{2}}{\sqrt[3]{198 p^{2}} {\sqrt[3]{198 p^{2}}}^{2}}}$

Step 2.1.2

Raise $\sqrt[3]{198 p^{2}}$ to the power of $1$ .

$\sqrt{\frac{99 {\sqrt[3]{198 p^{2}}}^{2}}{{\sqrt[3]{198 p^{2}}}^{1} {\sqrt[3]{198 p^{2}}}^{2}}}$

Step 2.1.3

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

$\sqrt{\frac{99 {\sqrt[3]{198 p^{2}}}^{2}}{{\sqrt[3]{198 p^{2}}}^{1 + 2}}}$

Step 2.1.4

Add $1$ and $2$ .

$\sqrt{\frac{99 {\sqrt[3]{198 p^{2}}}^{2}}{{\sqrt[3]{198 p^{2}}}^{3}}}$

Step 2.1.5

Rewrite ${\sqrt[3]{198 p^{2}}}^{3}$ as $198 p^{2}$ .

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Step 2.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{198 p^{2}}$ as ${(198 p^{2})}^{\frac{1}{3}}$ .

$\sqrt{\frac{99 {\sqrt[3]{198 p^{2}}}^{2}}{{({(198 p^{2})}^{\frac{1}{3}})}^{3}}}$

Step 2.1.5.2

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

$\sqrt{\frac{99 {\sqrt[3]{198 p^{2}}}^{2}}{{(198 p^{2})}^{\frac{1}{3} \cdot 3}}}$

Step 2.1.5.3

Combine $\frac{1}{3}$ and $3$ .

$\sqrt{\frac{99 {\sqrt[3]{198 p^{2}}}^{2}}{{(198 p^{2})}^{\frac{3}{3}}}}$

Step 2.1.5.4

Cancel the common factor of $3$ .

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Step 2.1.5.4.1

Cancel the common factor.

$\sqrt{\frac{99 {\sqrt[3]{198 p^{2}}}^{2}}{{(198 p^{2})}^{\frac{3}{3}}}}$

Step 2.1.5.4.2

Rewrite the expression.

$\sqrt{\frac{99 {\sqrt[3]{198 p^{2}}}^{2}}{{(198 p^{2})}^{1}}}$

Step 2.1.5.5

Simplify.

$\sqrt{\frac{99 {\sqrt[3]{198 p^{2}}}^{2}}{198 p^{2}}}$

Step 2.2

Cancel the common factor of $99$ and $198$ .

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Step 2.2.1

Factor $99$ out of $99 {\sqrt[3]{198 p^{2}}}^{2}$ .

$\sqrt{\frac{99 ({\sqrt[3]{198 p^{2}}}^{2})}{198 p^{2}}}$

Step 2.2.2

Cancel the common factors.

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Step 2.2.2.1

Factor $99$ out of $198 p^{2}$ .

$\sqrt{\frac{99 ({\sqrt[3]{198 p^{2}}}^{2})}{99 (2 p^{2})}}$

Step 2.2.2.2

Cancel the common factor.

$\sqrt{\frac{99 {\sqrt[3]{198 p^{2}}}^{2}}{99 (2 p^{2})}}$

Step 2.2.2.3

Rewrite the expression.

$\sqrt{\frac{{\sqrt[3]{198 p^{2}}}^{2}}{2 p^{2}}}$

Step 3

Simplify the numerator.

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Step 3.1

Rewrite ${\sqrt[3]{198 p^{2}}}^{2}$ as $\sqrt[3]{{(198 p^{2})}^{2}}$ .

$\sqrt{\frac{\sqrt[3]{{(198 p^{2})}^{2}}}{2 p^{2}}}$

Step 3.2

Apply the product rule to $198 p^{2}$ .

$\sqrt{\frac{\sqrt[3]{198^{2} {(p^{2})}^{2}}}{2 p^{2}}}$

Step 3.3

Raise $198$ to the power of $2$ .

$\sqrt{\frac{\sqrt[3]{39204 {(p^{2})}^{2}}}{2 p^{2}}}$

Step 3.4

Multiply the exponents in ${(p^{2})}^{2}$ .

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Step 3.4.1

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

$\sqrt{\frac{\sqrt[3]{39204 p^{2 \cdot 2}}}{2 p^{2}}}$

Step 3.4.2

Multiply $2$ by $2$ .

$\sqrt{\frac{\sqrt[3]{39204 p^{4}}}{2 p^{2}}}$

Step 3.5

Rewrite $39204 p^{4}$ as ${(3 p)}^{3} \cdot (1452 p)$ .

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Step 3.5.1

Factor $27$ out of $39204$ .

$\sqrt{\frac{\sqrt[3]{27 (1452) p^{4}}}{2 p^{2}}}$

Step 3.5.2

Rewrite $27$ as $3^{3}$ .

$\sqrt{\frac{\sqrt[3]{3^{3} \cdot 1452 p^{4}}}{2 p^{2}}}$

Step 3.5.3

Factor out $p^{3}$ .

$\sqrt{\frac{\sqrt[3]{3^{3} \cdot 1452 (p^{3} p)}}{2 p^{2}}}$

Step 3.5.4

Move $1452$ .

$\sqrt{\frac{\sqrt[3]{3^{3} p^{3} \cdot 1452 p}}{2 p^{2}}}$

Step 3.5.5

Rewrite $3^{3} p^{3}$ as ${(3 p)}^{3}$ .

$\sqrt{\frac{\sqrt[3]{{(3 p)}^{3} \cdot 1452 p}}{2 p^{2}}}$

Step 3.5.6

Add parentheses.

$\sqrt{\frac{\sqrt[3]{{(3 p)}^{3} \cdot (1452 p)}}{2 p^{2}}}$

Step 3.6

Pull terms out from under the radical.

$\sqrt{\frac{3 p \sqrt[3]{1452 p}}{2 p^{2}}}$

Step 4

Cancel the common factor of $p$ and $p^{2}$ .

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Step 4.1

Factor $p$ out of $3 p \sqrt[3]{1452 p}$ .

$\sqrt{\frac{p (3 \sqrt[3]{1452 p})}{2 p^{2}}}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $p$ out of $2 p^{2}$ .

$\sqrt{\frac{p (3 \sqrt[3]{1452 p})}{p (2 p)}}$

Step 4.2.2

Cancel the common factor.

$\sqrt{\frac{p (3 \sqrt[3]{1452 p})}{p (2 p)}}$

Step 4.2.3

Rewrite the expression.

$\sqrt{\frac{3 \sqrt[3]{1452 p}}{2 p}}$

Step 5

Rewrite $\sqrt{\frac{3 \sqrt[3]{1452 p}}{2 p}}$ as $\frac{\sqrt{3 \sqrt[3]{1452 p}}}{\sqrt{2 p}}$ .

$\frac{\sqrt{3 \sqrt[3]{1452 p}}}{\sqrt{2 p}}$

Step 6

Multiply $\frac{\sqrt{3 \sqrt[3]{1452 p}}}{\sqrt{2 p}}$ by $\frac{\sqrt{2 p}}{\sqrt{2 p}}$ .

$\frac{\sqrt{3 \sqrt[3]{1452 p}}}{\sqrt{2 p}} \cdot \frac{\sqrt{2 p}}{\sqrt{2 p}}$

Step 7

Combine and simplify the denominator.

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Step 7.1

Multiply $\frac{\sqrt{3 \sqrt[3]{1452 p}}}{\sqrt{2 p}}$ by $\frac{\sqrt{2 p}}{\sqrt{2 p}}$ .

$\frac{\sqrt{3 \sqrt[3]{1452 p}} \sqrt{2 p}}{\sqrt{2 p} \sqrt{2 p}}$

Step 7.2

Raise $\sqrt{2 p}$ to the power of $1$ .

$\frac{\sqrt{3 \sqrt[3]{1452 p}} \sqrt{2 p}}{{\sqrt{2 p}}^{1} \sqrt{2 p}}$

Step 7.3

Raise $\sqrt{2 p}$ to the power of $1$ .

$\frac{\sqrt{3 \sqrt[3]{1452 p}} \sqrt{2 p}}{{\sqrt{2 p}}^{1} {\sqrt{2 p}}^{1}}$

Step 7.4

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

$\frac{\sqrt{3 \sqrt[3]{1452 p}} \sqrt{2 p}}{{\sqrt{2 p}}^{1 + 1}}$

Step 7.5

Add $1$ and $1$ .

$\frac{\sqrt{3 \sqrt[3]{1452 p}} \sqrt{2 p}}{{\sqrt{2 p}}^{2}}$

Step 7.6

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

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Step 7.6.1

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

$\frac{\sqrt{3 \sqrt[3]{1452 p}} \sqrt{2 p}}{{({(2 p)}^{\frac{1}{2}})}^{2}}$

Step 7.6.2

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

$\frac{\sqrt{3 \sqrt[3]{1452 p}} \sqrt{2 p}}{{(2 p)}^{\frac{1}{2} \cdot 2}}$

Step 7.6.3

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

$\frac{\sqrt{3 \sqrt[3]{1452 p}} \sqrt{2 p}}{{(2 p)}^{\frac{2}{2}}}$

Step 7.6.4

Cancel the common factor of $2$ .

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Step 7.6.4.1

Cancel the common factor.

$\frac{\sqrt{3 \sqrt[3]{1452 p}} \sqrt{2 p}}{{(2 p)}^{\frac{2}{2}}}$

Step 7.6.4.2

Rewrite the expression.

$\frac{\sqrt{3 \sqrt[3]{1452 p}} \sqrt{2 p}}{{(2 p)}^{1}}$

Step 7.6.5

Simplify.

$\frac{\sqrt{3 \sqrt[3]{1452 p}} \sqrt{2 p}}{2 p}$

Step 8

Simplify the numerator.

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Step 8.1

Combine using the product rule for radicals.

$\frac{\sqrt{3 \sqrt[3]{1452 p} (2 p)}}{2 p}$

Step 8.2

Multiply $2$ by $3$ .

$\frac{\sqrt{6 \sqrt[3]{1452 p} p}}{2 p}$