Basic Math Examples

$\frac{\sqrt{27}}{\sqrt{63}}$

Step 1

Combine

$\sqrt{27}$ and

$\sqrt{63}$ into a single radical.

$\sqrt{\frac{27}{63}}$

Step 2

Cancel the common factor of

$27$ and

$63$ .

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Step 2.1

Factor

$9$ out of

$27$ .

$\sqrt{\frac{9 (3)}{63}}$

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor

$9$ out of

$63$ .

$\sqrt{\frac{9 \cdot 3}{9 \cdot 7}}$

Step 2.2.2

Cancel the common factor.

$\sqrt{\frac{9 \cdot 3}{9 \cdot 7}}$

Step 2.2.3

Rewrite the expression.

$\sqrt{\frac{3}{7}}$

Step 3

Rewrite

$\sqrt{\frac{3}{7}}$ as

$\frac{\sqrt{3}}{\sqrt{7}}$ .

$\frac{\sqrt{3}}{\sqrt{7}}$

Step 4

Multiply

$\frac{\sqrt{3}}{\sqrt{7}}$ by

$\frac{\sqrt{7}}{\sqrt{7}}$ .

$\frac{\sqrt{3}}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}$

Step 5

Combine and simplify the denominator.

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Step 5.1

Multiply

$\frac{\sqrt{3}}{\sqrt{7}}$ by

$\frac{\sqrt{7}}{\sqrt{7}}$ .

$\frac{\sqrt{3} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 5.2

Raise

$\sqrt{7}$ to the power of

$1$ .

$\frac{\sqrt{3} \sqrt{7}}{{\sqrt{7}}^{1} \sqrt{7}}$

Step 5.3

Raise

$\sqrt{7}$ to the power of

$1$ .

$\frac{\sqrt{3} \sqrt{7}}{{\sqrt{7}}^{1} {\sqrt{7}}^{1}}$

Step 5.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sqrt{3} \sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Step 5.5

Add

$1$ and

$1$ .

$\frac{\sqrt{3} \sqrt{7}}{{\sqrt{7}}^{2}}$

Step 5.6

Rewrite

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

$7$ .

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Step 5.6.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{7}$ as

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

$\frac{\sqrt{3} \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

Step 5.6.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$\frac{\sqrt{3} \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

Step 5.6.3

Combine

$\frac{1}{2}$ and

$2$ .

$\frac{\sqrt{3} \sqrt{7}}{7^{\frac{2}{2}}}$

Step 5.6.4

Cancel the common factor of

$2$ .

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Step 5.6.4.1

Cancel the common factor.

$\frac{\sqrt{3} \sqrt{7}}{7^{\frac{2}{2}}}$

Step 5.6.4.2

Rewrite the expression.

$\frac{\sqrt{3} \sqrt{7}}{7^{1}}$

Step 5.6.5

Evaluate the exponent.

$\frac{\sqrt{3} \sqrt{7}}{7}$

Step 6

Simplify the numerator.

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Step 6.1

Combine using the product rule for radicals.

$\frac{\sqrt{3 \cdot 7}}{7}$

Step 6.2

Multiply

$3$ by

$7$ .

$\frac{\sqrt{21}}{7}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{21}}{7}$

Decimal Form:

$0.65465367 \dots$