Finite Math Examples

$\frac{1}{4} + \sqrt{\frac{13}{2}}$

Step 1

Simplify each term.

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Step 1.1

Rewrite $\sqrt{\frac{13}{2}}$ as $\frac{\sqrt{13}}{\sqrt{2}}$ .

$\frac{1}{4} + \frac{\sqrt{13}}{\sqrt{2}}$

Step 1.2

Multiply $\frac{\sqrt{13}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{1}{4} + \frac{\sqrt{13}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 1.3

Combine and simplify the denominator.

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Step 1.3.1

Multiply $\frac{\sqrt{13}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{1}{4} + \frac{\sqrt{13} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 1.3.2

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

$\frac{1}{4} + \frac{\sqrt{13} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 1.3.3

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

$\frac{1}{4} + \frac{\sqrt{13} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 1.3.4

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

$\frac{1}{4} + \frac{\sqrt{13} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 1.3.5

Add $1$ and $1$ .

$\frac{1}{4} + \frac{\sqrt{13} \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 1.3.6

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

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Step 1.3.6.1

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

$\frac{1}{4} + \frac{\sqrt{13} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 1.3.6.2

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

$\frac{1}{4} + \frac{\sqrt{13} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 1.3.6.3

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

$\frac{1}{4} + \frac{\sqrt{13} \sqrt{2}}{2^{\frac{2}{2}}}$

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.

$\frac{1}{4} + \frac{\sqrt{13} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 1.3.6.4.2

Rewrite the expression.

$\frac{1}{4} + \frac{\sqrt{13} \sqrt{2}}{2^{1}}$

Step 1.3.6.5

Evaluate the exponent.

$\frac{1}{4} + \frac{\sqrt{13} \sqrt{2}}{2}$

Step 1.4

Simplify the numerator.

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Step 1.4.1

Combine using the product rule for radicals.

$\frac{1}{4} + \frac{\sqrt{13 \cdot 2}}{2}$

Step 1.4.2

Multiply $13$ by $2$ .

$\frac{1}{4} + \frac{\sqrt{26}}{2}$

Step 2

To write $\frac{\sqrt{26}}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{4} + \frac{\sqrt{26}}{2} \cdot \frac{2}{2}$

Step 3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{\sqrt{26}}{2}$ by $\frac{2}{2}$ .

$\frac{1}{4} + \frac{\sqrt{26} \cdot 2}{2 \cdot 2}$

Step 3.2

Multiply $2$ by $2$ .

$\frac{1}{4} + \frac{\sqrt{26} \cdot 2}{4}$

Step 4

Combine the numerators over the common denominator.

$\frac{1 + \sqrt{26} \cdot 2}{4}$

Step 5

Move $2$ to the left of $\sqrt{26}$ .

$\frac{1 + 2 \sqrt{26}}{4}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{1 + 2 \sqrt{26}}{4}$

Decimal Form:

$2.79950975 \dots$