Calculus Examples

$7 (\frac{8}{3}) \sqrt{4 - \frac{8}{3}}$

Step 1

Multiply $7 (\frac{8}{3})$ .

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Step 1.1

Combine $7$ and $\frac{8}{3}$ .

$\frac{7 \cdot 8}{3} \cdot \sqrt{4 - \frac{8}{3}}$

Step 1.2

Multiply $7$ by $8$ .

$\frac{56}{3} \cdot \sqrt{4 - \frac{8}{3}}$

Step 2

To write $4$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{56}{3} \cdot \sqrt{4 \cdot \frac{3}{3} - \frac{8}{3}}$

Step 3

Combine $4$ and $\frac{3}{3}$ .

$\frac{56}{3} \cdot \sqrt{\frac{4 \cdot 3}{3} - \frac{8}{3}}$

Step 4

Combine the numerators over the common denominator.

$\frac{56}{3} \cdot \sqrt{\frac{4 \cdot 3 - 8}{3}}$

Step 5

Simplify the numerator.

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Step 5.1

Multiply $4$ by $3$ .

$\frac{56}{3} \cdot \sqrt{\frac{12 - 8}{3}}$

Step 5.2

Subtract $8$ from $12$ .

$\frac{56}{3} \cdot \sqrt{\frac{4}{3}}$

Step 6

Rewrite $\sqrt{\frac{4}{3}}$ as $\frac{\sqrt{4}}{\sqrt{3}}$ .

$\frac{56}{3} \cdot \frac{\sqrt{4}}{\sqrt{3}}$

Step 7

Simplify the numerator.

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Step 7.1

Rewrite $4$ as $2^{2}$ .

$\frac{56}{3} \cdot \frac{\sqrt{2^{2}}}{\sqrt{3}}$

Step 7.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{56}{3} \cdot \frac{2}{\sqrt{3}}$

Step 8

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

$\frac{56}{3} \cdot (\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 9

Combine and simplify the denominator.

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Step 9.1

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

$\frac{56}{3} \cdot \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 9.2

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

$\frac{56}{3} \cdot \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 9.3

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

$\frac{56}{3} \cdot \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 9.4

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

$\frac{56}{3} \cdot \frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 9.5

Add $1$ and $1$ .

$\frac{56}{3} \cdot \frac{2 \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 9.6

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

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Step 9.6.1

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

$\frac{56}{3} \cdot \frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 9.6.2

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

$\frac{56}{3} \cdot \frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 9.6.3

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

$\frac{56}{3} \cdot \frac{2 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 9.6.4

Cancel the common factor of $2$ .

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Step 9.6.4.1

Cancel the common factor.

$\frac{56}{3} \cdot \frac{2 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 9.6.4.2

Rewrite the expression.

$\frac{56}{3} \cdot \frac{2 \sqrt{3}}{3^{1}}$

Step 9.6.5

Evaluate the exponent.

$\frac{56}{3} \cdot \frac{2 \sqrt{3}}{3}$

Step 10

Multiply $\frac{56}{3} \cdot \frac{2 \sqrt{3}}{3}$ .

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Step 10.1

Multiply $\frac{56}{3}$ by $\frac{2 \sqrt{3}}{3}$ .

$\frac{56 (2 \sqrt{3})}{3 \cdot 3}$

Step 10.2

Multiply $2$ by $56$ .

$\frac{112 \sqrt{3}}{3 \cdot 3}$

Step 10.3

Multiply $3$ by $3$ .

$\frac{112 \sqrt{3}}{9}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{112 \sqrt{3}}{9}$

Decimal Form:

$21.55441004 \dots$