Calculus Examples

$\frac{\arctan (\sqrt{\frac{253}{3}}) - \arctan (\sqrt{\frac{13}{3}})}{\sqrt{3}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $\sqrt{\frac{253}{3}}$ as $\frac{\sqrt{253}}{\sqrt{3}}$ .

$\frac{\arctan (\frac{\sqrt{253}}{\sqrt{3}}) - \arctan (\sqrt{\frac{13}{3}})}{\sqrt{3}}$

Step 1.2

Multiply $\frac{\sqrt{253}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{\arctan (\frac{\sqrt{253}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}) - \arctan (\sqrt{\frac{13}{3}})}{\sqrt{3}}$

Step 1.3

Combine and simplify the denominator.

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Step 1.3.1

Multiply $\frac{\sqrt{253}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{\arctan (\frac{\sqrt{253} \sqrt{3}}{\sqrt{3} \sqrt{3}}) - \arctan (\sqrt{\frac{13}{3}})}{\sqrt{3}}$

Step 1.3.2

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{\arctan (\frac{\sqrt{253} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}) - \arctan (\sqrt{\frac{13}{3}})}{\sqrt{3}}$

Step 1.3.3

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{\arctan (\frac{\sqrt{253} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}) - \arctan (\sqrt{\frac{13}{3}})}{\sqrt{3}}$

Step 1.3.4

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

$\frac{\arctan (\frac{\sqrt{253} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}) - \arctan (\sqrt{\frac{13}{3}})}{\sqrt{3}}$

Step 1.3.5

Add $1$ and $1$ .

$\frac{\arctan (\frac{\sqrt{253} \sqrt{3}}{{\sqrt{3}}^{2}}) - \arctan (\sqrt{\frac{13}{3}})}{\sqrt{3}}$

Step 1.3.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 1.3.6.1

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

$\frac{\arctan (\frac{\sqrt{253} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}) - \arctan (\sqrt{\frac{13}{3}})}{\sqrt{3}}$

Step 1.3.6.2

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

$\frac{\arctan (\frac{\sqrt{253} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}) - \arctan (\sqrt{\frac{13}{3}})}{\sqrt{3}}$

Step 1.3.6.3

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

$\frac{\arctan (\frac{\sqrt{253} \sqrt{3}}{3^{\frac{2}{2}}}) - \arctan (\sqrt{\frac{13}{3}})}{\sqrt{3}}$

Step 1.3.6.4

Cancel the common factor of $2$ .

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Step 1.3.6.4.1

Cancel the common factor.

$\frac{\arctan (\frac{\sqrt{253} \sqrt{3}}{3^{\frac{2}{2}}}) - \arctan (\sqrt{\frac{13}{3}})}{\sqrt{3}}$

Step 1.3.6.4.2

Rewrite the expression.

$\frac{\arctan (\frac{\sqrt{253} \sqrt{3}}{3^{1}}) - \arctan (\sqrt{\frac{13}{3}})}{\sqrt{3}}$

Step 1.3.6.5

Evaluate the exponent.

$\frac{\arctan (\frac{\sqrt{253} \sqrt{3}}{3}) - \arctan (\sqrt{\frac{13}{3}})}{\sqrt{3}}$

Step 1.4

Simplify the numerator.

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Step 1.4.1

Combine using the product rule for radicals.

$\frac{\arctan (\frac{\sqrt{253 \cdot 3}}{3}) - \arctan (\sqrt{\frac{13}{3}})}{\sqrt{3}}$

Step 1.4.2

Multiply $253$ by $3$ .

$\frac{\arctan (\frac{\sqrt{759}}{3}) - \arctan (\sqrt{\frac{13}{3}})}{\sqrt{3}}$

Step 1.5

Rewrite $\sqrt{\frac{13}{3}}$ as $\frac{\sqrt{13}}{\sqrt{3}}$ .

$\frac{\arctan (\frac{\sqrt{759}}{3}) - \arctan (\frac{\sqrt{13}}{\sqrt{3}})}{\sqrt{3}}$

Step 1.6

Multiply $\frac{\sqrt{13}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{\arctan (\frac{\sqrt{759}}{3}) - \arctan (\frac{\sqrt{13}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})}{\sqrt{3}}$

Step 1.7

Combine and simplify the denominator.

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Step 1.7.1

Multiply $\frac{\sqrt{13}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{\arctan (\frac{\sqrt{759}}{3}) - \arctan (\frac{\sqrt{13} \sqrt{3}}{\sqrt{3} \sqrt{3}})}{\sqrt{3}}$

Step 1.7.2

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{\arctan (\frac{\sqrt{759}}{3}) - \arctan (\frac{\sqrt{13} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}})}{\sqrt{3}}$

Step 1.7.3

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{\arctan (\frac{\sqrt{759}}{3}) - \arctan (\frac{\sqrt{13} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}})}{\sqrt{3}}$

Step 1.7.4

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

$\frac{\arctan (\frac{\sqrt{759}}{3}) - \arctan (\frac{\sqrt{13} \sqrt{3}}{{\sqrt{3}}^{1 + 1}})}{\sqrt{3}}$

Step 1.7.5

Add $1$ and $1$ .

$\frac{\arctan (\frac{\sqrt{759}}{3}) - \arctan (\frac{\sqrt{13} \sqrt{3}}{{\sqrt{3}}^{2}})}{\sqrt{3}}$

Step 1.7.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 1.7.6.1

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

$\frac{\arctan (\frac{\sqrt{759}}{3}) - \arctan (\frac{\sqrt{13} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}})}{\sqrt{3}}$

Step 1.7.6.2

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

$\frac{\arctan (\frac{\sqrt{759}}{3}) - \arctan (\frac{\sqrt{13} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}})}{\sqrt{3}}$

Step 1.7.6.3

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

$\frac{\arctan (\frac{\sqrt{759}}{3}) - \arctan (\frac{\sqrt{13} \sqrt{3}}{3^{\frac{2}{2}}})}{\sqrt{3}}$

Step 1.7.6.4

Cancel the common factor of $2$ .

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Step 1.7.6.4.1

Cancel the common factor.

$\frac{\arctan (\frac{\sqrt{759}}{3}) - \arctan (\frac{\sqrt{13} \sqrt{3}}{3^{\frac{2}{2}}})}{\sqrt{3}}$

Step 1.7.6.4.2

Rewrite the expression.

$\frac{\arctan (\frac{\sqrt{759}}{3}) - \arctan (\frac{\sqrt{13} \sqrt{3}}{3^{1}})}{\sqrt{3}}$

Step 1.7.6.5

Evaluate the exponent.

$\frac{\arctan (\frac{\sqrt{759}}{3}) - \arctan (\frac{\sqrt{13} \sqrt{3}}{3})}{\sqrt{3}}$

Step 1.8

Simplify the numerator.

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Step 1.8.1

Combine using the product rule for radicals.

$\frac{\arctan (\frac{\sqrt{759}}{3}) - \arctan (\frac{\sqrt{13 \cdot 3}}{3})}{\sqrt{3}}$

Step 1.8.2

Multiply $13$ by $3$ .

$\frac{\arctan (\frac{\sqrt{759}}{3}) - \arctan (\frac{\sqrt{39}}{3})}{\sqrt{3}}$

Step 2

Multiply $\frac{\arctan (\frac{\sqrt{759}}{3}) - \arctan (\frac{\sqrt{39}}{3})}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{\arctan (\frac{\sqrt{759}}{3}) - \arctan (\frac{\sqrt{39}}{3})}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{\arctan (\frac{\sqrt{759}}{3}) - \arctan (\frac{\sqrt{39}}{3})}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{(\arctan (\frac{\sqrt{759}}{3}) - \arctan (\frac{\sqrt{39}}{3})) \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 3.2

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{(\arctan (\frac{\sqrt{759}}{3}) - \arctan (\frac{\sqrt{39}}{3})) \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 3.3

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{(\arctan (\frac{\sqrt{759}}{3}) - \arctan (\frac{\sqrt{39}}{3})) \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 3.4

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

$\frac{(\arctan (\frac{\sqrt{759}}{3}) - \arctan (\frac{\sqrt{39}}{3})) \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 3.5

Add $1$ and $1$ .

$\frac{(\arctan (\frac{\sqrt{759}}{3}) - \arctan (\frac{\sqrt{39}}{3})) \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 3.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 3.6.1

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

$\frac{(\arctan (\frac{\sqrt{759}}{3}) - \arctan (\frac{\sqrt{39}}{3})) \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 3.6.2

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

$\frac{(\arctan (\frac{\sqrt{759}}{3}) - \arctan (\frac{\sqrt{39}}{3})) \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 3.6.3

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

$\frac{(\arctan (\frac{\sqrt{759}}{3}) - \arctan (\frac{\sqrt{39}}{3})) \sqrt{3}}{3^{\frac{2}{2}}}$

Step 3.6.4

Cancel the common factor of $2$ .

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Step 3.6.4.1

Cancel the common factor.

$\frac{(\arctan (\frac{\sqrt{759}}{3}) - \arctan (\frac{\sqrt{39}}{3})) \sqrt{3}}{3^{\frac{2}{2}}}$

Step 3.6.4.2

Rewrite the expression.

$\frac{(\arctan (\frac{\sqrt{759}}{3}) - \arctan (\frac{\sqrt{39}}{3})) \sqrt{3}}{3^{1}}$

Step 3.6.5

Evaluate the exponent.

$\frac{(\arctan (\frac{\sqrt{759}}{3}) - \arctan (\frac{\sqrt{39}}{3})) \sqrt{3}}{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{(\arctan (\frac{\sqrt{759}}{3}) - \arctan (\frac{\sqrt{39}}{3})) \sqrt{3}}{3}$

Decimal Form:

$0.19593343 \dots$