Calculus Examples

$\sqrt{\frac{23}{2} - 8}$

Step 1

To write $- 8$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\sqrt{\frac{23}{2} - 8 \cdot \frac{2}{2}}$

Step 2

Combine $- 8$ and $\frac{2}{2}$ .

$\sqrt{\frac{23}{2} + \frac{- 8 \cdot 2}{2}}$

Step 3

Combine the numerators over the common denominator.

$\sqrt{\frac{23 - 8 \cdot 2}{2}}$

Step 4

Simplify the numerator.

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Step 4.1

Multiply $- 8$ by $2$ .

$\sqrt{\frac{23 - 16}{2}}$

Step 4.2

Subtract $16$ from $23$ .

$\sqrt{\frac{7}{2}}$

Step 5

Rewrite $\sqrt{\frac{7}{2}}$ as $\frac{\sqrt{7}}{\sqrt{2}}$ .

$\frac{\sqrt{7}}{\sqrt{2}}$

Step 6

Multiply $\frac{\sqrt{7}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\sqrt{7}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 7

Combine and simplify the denominator.

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Step 7.1

Multiply $\frac{\sqrt{7}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\sqrt{7} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 7.2

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

$\frac{\sqrt{7} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 7.3

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

$\frac{\sqrt{7} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 7.4

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

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

Step 7.5

Add $1$ and $1$ .

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

Step 7.6

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

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Step 7.6.1

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

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

Step 7.6.2

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

$\frac{\sqrt{7} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 7.6.3

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

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

Step 7.6.4

Cancel the common factor of $2$ .

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Step 7.6.4.1

Cancel the common factor.

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

Step 7.6.4.2

Rewrite the expression.

$\frac{\sqrt{7} \sqrt{2}}{2^{1}}$

Step 7.6.5

Evaluate the exponent.

$\frac{\sqrt{7} \sqrt{2}}{2}$

Step 8

Simplify the numerator.

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Step 8.1

Combine using the product rule for radicals.

$\frac{\sqrt{7 \cdot 2}}{2}$

Step 8.2

Multiply $7$ by $2$ .

$\frac{\sqrt{14}}{2}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{14}}{2}$

Decimal Form:

$1.87082869 \dots$