Basic Math Examples

$2 \sqrt{28} - 7 \sqrt{\frac{2}{7}} - \sqrt{63}$

Step 1

Simplify each term.

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Step 1.1

Rewrite $28$ as $2^{2} \cdot 7$ .

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Step 1.1.1

Factor $4$ out of $28$ .

$2 \sqrt{4 (7)} - 7 \sqrt{\frac{2}{7}} - \sqrt{63}$

Step 1.1.2

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

$2 \sqrt{2^{2} \cdot 7} - 7 \sqrt{\frac{2}{7}} - \sqrt{63}$

Step 1.2

Pull terms out from under the radical.

$2 (2 \sqrt{7}) - 7 \sqrt{\frac{2}{7}} - \sqrt{63}$

Step 1.3

Multiply $2$ by $2$ .

$4 \sqrt{7} - 7 \sqrt{\frac{2}{7}} - \sqrt{63}$

Step 1.4

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

$4 \sqrt{7} - 7 \frac{\sqrt{2}}{\sqrt{7}} - \sqrt{63}$

Step 1.5

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

$4 \sqrt{7} - 7 (\frac{\sqrt{2}}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}) - \sqrt{63}$

Step 1.6

Combine and simplify the denominator.

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Step 1.6.1

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

$4 \sqrt{7} - 7 \frac{\sqrt{2} \sqrt{7}}{\sqrt{7} \sqrt{7}} - \sqrt{63}$

Step 1.6.2

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

$4 \sqrt{7} - 7 \frac{\sqrt{2} \sqrt{7}}{{\sqrt{7}}^{1} \sqrt{7}} - \sqrt{63}$

Step 1.6.3

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

$4 \sqrt{7} - 7 \frac{\sqrt{2} \sqrt{7}}{{\sqrt{7}}^{1} {\sqrt{7}}^{1}} - \sqrt{63}$

Step 1.6.4

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

$4 \sqrt{7} - 7 \frac{\sqrt{2} \sqrt{7}}{{\sqrt{7}}^{1 + 1}} - \sqrt{63}$

Step 1.6.5

Add $1$ and $1$ .

$4 \sqrt{7} - 7 \frac{\sqrt{2} \sqrt{7}}{{\sqrt{7}}^{2}} - \sqrt{63}$

Step 1.6.6

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

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Step 1.6.6.1

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

$4 \sqrt{7} - 7 \frac{\sqrt{2} \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}} - \sqrt{63}$

Step 1.6.6.2

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

$4 \sqrt{7} - 7 \frac{\sqrt{2} \sqrt{7}}{7^{\frac{1}{2} \cdot 2}} - \sqrt{63}$

Step 1.6.6.3

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

$4 \sqrt{7} - 7 \frac{\sqrt{2} \sqrt{7}}{7^{\frac{2}{2}}} - \sqrt{63}$

Step 1.6.6.4

Cancel the common factor of $2$ .

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Step 1.6.6.4.1

Cancel the common factor.

$4 \sqrt{7} - 7 \frac{\sqrt{2} \sqrt{7}}{7^{\frac{2}{2}}} - \sqrt{63}$

Step 1.6.6.4.2

Rewrite the expression.

$4 \sqrt{7} - 7 \frac{\sqrt{2} \sqrt{7}}{7^{1}} - \sqrt{63}$

Step 1.6.6.5

Evaluate the exponent.

$4 \sqrt{7} - 7 \frac{\sqrt{2} \sqrt{7}}{7} - \sqrt{63}$

Step 1.7

Simplify the numerator.

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Step 1.7.1

Combine using the product rule for radicals.

$4 \sqrt{7} - 7 \frac{\sqrt{2 \cdot 7}}{7} - \sqrt{63}$

Step 1.7.2

Multiply $2$ by $7$ .

$4 \sqrt{7} - 7 \frac{\sqrt{14}}{7} - \sqrt{63}$

Step 1.8

Cancel the common factor of $7$ .

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Step 1.8.1

Factor $7$ out of $- 7$ .

$4 \sqrt{7} + 7 (- 1) \frac{\sqrt{14}}{7} - \sqrt{63}$

Step 1.8.2

Cancel the common factor.

$4 \sqrt{7} + 7 \cdot - 1 \frac{\sqrt{14}}{7} - \sqrt{63}$

Step 1.8.3

Rewrite the expression.

$4 \sqrt{7} - 1 \sqrt{14} - \sqrt{63}$

Step 1.9

Rewrite $- 1 \sqrt{14}$ as $- \sqrt{14}$ .

$4 \sqrt{7} - \sqrt{14} - \sqrt{63}$

Step 1.10

Rewrite $63$ as $3^{2} \cdot 7$ .

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Step 1.10.1

Factor $9$ out of $63$ .

$4 \sqrt{7} - \sqrt{14} - \sqrt{9 (7)}$

Step 1.10.2

Rewrite $9$ as $3^{2}$ .

$4 \sqrt{7} - \sqrt{14} - \sqrt{3^{2} \cdot 7}$

Step 1.11

Pull terms out from under the radical.

$4 \sqrt{7} - \sqrt{14} - (3 \sqrt{7})$

Step 1.12

Multiply $3$ by $- 1$ .

$4 \sqrt{7} - \sqrt{14} - 3 \sqrt{7}$

Step 2

Subtract $3 \sqrt{7}$ from $4 \sqrt{7}$ .

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

Step 3

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 1.09590607 \dots$