Calculus Examples

$2736 {(\frac{2}{\sqrt{3}})}^{7}$

Step 1

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$2736 {(\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})}^{7}$

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$2736 {(\frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}})}^{7}$

Step 2.2

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

$2736 {(\frac{2 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}})}^{7}$

Step 2.3

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

$2736 {(\frac{2 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}})}^{7}$

Step 2.4

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

$2736 {(\frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}})}^{7}$

Step 2.5

Add $1$ and $1$ .

$2736 {(\frac{2 \sqrt{3}}{{\sqrt{3}}^{2}})}^{7}$

Step 2.6

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

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Step 2.6.1

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

$2736 {(\frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}})}^{7}$

Step 2.6.2

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

$2736 {(\frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}})}^{7}$

Step 2.6.3

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

$2736 {(\frac{2 \sqrt{3}}{3^{\frac{2}{2}}})}^{7}$

Step 2.6.4

Cancel the common factor of $2$ .

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Step 2.6.4.1

Cancel the common factor.

$2736 {(\frac{2 \sqrt{3}}{3^{\frac{2}{2}}})}^{7}$

Step 2.6.4.2

Rewrite the expression.

$2736 {(\frac{2 \sqrt{3}}{3^{1}})}^{7}$

Step 2.6.5

Evaluate the exponent.

$2736 {(\frac{2 \sqrt{3}}{3})}^{7}$

Step 3

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.1

Apply the product rule to $\frac{2 \sqrt{3}}{3}$ .

$2736 \frac{{(2 \sqrt{3})}^{7}}{3^{7}}$

Step 3.2

Apply the product rule to $2 \sqrt{3}$ .

$2736 \frac{2^{7} {\sqrt{3}}^{7}}{3^{7}}$

Step 4

Simplify the numerator.

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Step 4.1

Raise $2$ to the power of $7$ .

$2736 \frac{128 {\sqrt{3}}^{7}}{3^{7}}$

Step 4.2

Rewrite ${\sqrt{3}}^{7}$ as $\sqrt{3^{7}}$ .

$2736 \frac{128 \sqrt{3^{7}}}{3^{7}}$

Step 4.3

Raise $3$ to the power of $7$ .

$2736 \frac{128 \sqrt{2187}}{3^{7}}$

Step 4.4

Rewrite $2187$ as $27^{2} \cdot 3$ .

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Step 4.4.1

Factor $729$ out of $2187$ .

$2736 \frac{128 \sqrt{729 (3)}}{3^{7}}$

Step 4.4.2

Rewrite $729$ as $27^{2}$ .

$2736 \frac{128 \sqrt{27^{2} \cdot 3}}{3^{7}}$

Step 4.5

Pull terms out from under the radical.

$2736 \frac{128 \cdot 27 \sqrt{3}}{3^{7}}$

Step 4.6

Multiply $128$ by $27$ .

$2736 \frac{3456 \sqrt{3}}{3^{7}}$

Step 5

Simplify terms.

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Step 5.1

Raise $3$ to the power of $7$ .

$2736 \frac{3456 \sqrt{3}}{2187}$

Step 5.2

Cancel the common factor of $9$ .

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Step 5.2.1

Factor $9$ out of $2736$ .

$9 (304) \frac{3456 \sqrt{3}}{2187}$

Step 5.2.2

Factor $9$ out of $2187$ .

$9 \cdot 304 \frac{3456 \sqrt{3}}{9 \cdot 243}$

Step 5.2.3

Cancel the common factor.

$9 \cdot 304 \frac{3456 \sqrt{3}}{9 \cdot 243}$

Step 5.2.4

Rewrite the expression.

$304 \frac{3456 \sqrt{3}}{243}$

Step 5.3

Combine $304$ and $\frac{3456 \sqrt{3}}{243}$ .

$\frac{304 (3456 \sqrt{3})}{243}$

Step 5.4

Multiply $3456$ by $304$ .

$\frac{1050624 \sqrt{3}}{243}$

Step 5.5

Cancel the common factor of $1050624$ and $243$ .

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Step 5.5.1

Factor $27$ out of $1050624 \sqrt{3}$ .

$\frac{27 (38912 \sqrt{3})}{243}$

Step 5.5.2

Cancel the common factors.

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Step 5.5.2.1

Factor $27$ out of $243$ .

$\frac{27 (38912 \sqrt{3})}{27 (9)}$

Step 5.5.2.2

Cancel the common factor.

$\frac{27 (38912 \sqrt{3})}{27 \cdot 9}$

Step 5.5.2.3

Rewrite the expression.

$\frac{38912 \sqrt{3}}{9}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{38912 \sqrt{3}}{9}$

Decimal Form:

$7488.61789156 \dots$