Algebra Examples

$\sqrt{14} - \sqrt{\frac{2}{7}}$

Step 1

Simplify each term.

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Step 1.1

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

$\sqrt{14} - \frac{\sqrt{2}}{\sqrt{7}}$

Step 1.2

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

$\sqrt{14} - (\frac{\sqrt{2}}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}})$

Step 1.3

Combine and simplify the denominator.

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Step 1.3.1

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

$\sqrt{14} - \frac{\sqrt{2} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 1.3.2

Raise $\sqrt{7}$ to the power of $1$ .

$\sqrt{14} - \frac{\sqrt{2} \sqrt{7}}{{\sqrt{7}}^{1} \sqrt{7}}$

Step 1.3.3

Raise $\sqrt{7}$ to the power of $1$ .

$\sqrt{14} - \frac{\sqrt{2} \sqrt{7}}{{\sqrt{7}}^{1} {\sqrt{7}}^{1}}$

Step 1.3.4

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

$\sqrt{14} - \frac{\sqrt{2} \sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Step 1.3.5

Add $1$ and $1$ .

$\sqrt{14} - \frac{\sqrt{2} \sqrt{7}}{{\sqrt{7}}^{2}}$

Step 1.3.6

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

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Step 1.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\sqrt{14} - \frac{\sqrt{2} \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

Step 1.3.6.2

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

$\sqrt{14} - \frac{\sqrt{2} \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

Step 1.3.6.3

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

$\sqrt{14} - \frac{\sqrt{2} \sqrt{7}}{7^{\frac{2}{2}}}$

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.

$\sqrt{14} - \frac{\sqrt{2} \sqrt{7}}{7^{\frac{2}{2}}}$

Step 1.3.6.4.2

Rewrite the expression.

$\sqrt{14} - \frac{\sqrt{2} \sqrt{7}}{7^{1}}$

Step 1.3.6.5

Evaluate the exponent.

$\sqrt{14} - \frac{\sqrt{2} \sqrt{7}}{7}$

Step 1.4

Simplify the numerator.

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Step 1.4.1

Combine using the product rule for radicals.

$\sqrt{14} - \frac{\sqrt{2 \cdot 7}}{7}$

Step 1.4.2

Multiply $2$ by $7$ .

$\sqrt{14} - \frac{\sqrt{14}}{7}$

Step 2

To write $\sqrt{14}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$\sqrt{14} \cdot \frac{7}{7} - \frac{\sqrt{14}}{7}$

Step 3

Combine fractions.

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Step 3.1

Combine $\sqrt{14}$ and $\frac{7}{7}$ .

$\frac{\sqrt{14} \cdot 7}{7} - \frac{\sqrt{14}}{7}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{\sqrt{14} \cdot 7 - \sqrt{14}}{7}$

Step 4

Simplify the numerator.

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Step 4.1

Move $7$ to the left of $\sqrt{14}$ .

$\frac{7 \cdot \sqrt{14} - \sqrt{14}}{7}$

Step 4.2

Subtract $\sqrt{14}$ from $7 \sqrt{14}$ .

$\frac{6 \sqrt{14}}{7}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{6 \sqrt{14}}{7}$

Decimal Form:

$3.20713490 \dots$