Pre-Algebra Examples

$\frac{\sqrt{39} \sqrt{39}}{\sqrt{13} \sqrt{98}}$

Step 1

Combine $\sqrt{39}$ and $\sqrt{13}$ into a single radical.

$\frac{\sqrt{\frac{39}{13}} \sqrt{39}}{\sqrt{98}}$

Step 2

Cancel the common factor of $39$ and $13$ .

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Step 2.1

Factor $13$ out of $39$ .

$\frac{\sqrt{\frac{13 \cdot 3}{13}} \sqrt{39}}{\sqrt{98}}$

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor $13$ out of $13$ .

$\frac{\sqrt{\frac{13 \cdot 3}{13 (1)}} \sqrt{39}}{\sqrt{98}}$

Step 2.2.2

Cancel the common factor.

$\frac{\sqrt{\frac{13 \cdot 3}{13 \cdot 1}} \sqrt{39}}{\sqrt{98}}$

Step 2.2.3

Rewrite the expression.

$\frac{\sqrt{\frac{3}{1}} \sqrt{39}}{\sqrt{98}}$

Step 2.2.4

Divide $3$ by $1$ .

$\frac{\sqrt{3} \sqrt{39}}{\sqrt{98}}$

Step 3

Simplify the numerator.

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Step 3.1

Combine using the product rule for radicals.

$\frac{\sqrt{3 \cdot 39}}{\sqrt{98}}$

Step 3.2

Multiply $3$ by $39$ .

$\frac{\sqrt{117}}{\sqrt{98}}$

Step 4

Simplify the denominator.

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Step 4.1

Rewrite $98$ as $7^{2} \cdot 2$ .

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Step 4.1.1

Factor $49$ out of $98$ .

$\frac{\sqrt{117}}{\sqrt{49 (2)}}$

Step 4.1.2

Rewrite $49$ as $7^{2}$ .

$\frac{\sqrt{117}}{\sqrt{7^{2} \cdot 2}}$

Step 4.2

Pull terms out from under the radical.

$\frac{\sqrt{117}}{7 \sqrt{2}}$

Step 5

Simplify the numerator.

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Step 5.1

Rewrite $117$ as $3^{2} \cdot 13$ .

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Step 5.1.1

Factor $9$ out of $117$ .

$\frac{\sqrt{9 (13)}}{7 \sqrt{2}}$

Step 5.1.2

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

$\frac{\sqrt{3^{2} \cdot 13}}{7 \sqrt{2}}$

Step 5.2

Pull terms out from under the radical.

$\frac{3 \sqrt{13}}{7 \sqrt{2}}$

Step 6

Multiply $\frac{3 \sqrt{13}}{7 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{3 \sqrt{13}}{7 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 7

Combine and simplify the denominator.

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Step 7.1

Multiply $\frac{3 \sqrt{13}}{7 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{3 \sqrt{13} \sqrt{2}}{7 \sqrt{2} \sqrt{2}}$

Step 7.2

Move $\sqrt{2}$ .

$\frac{3 \sqrt{13} \sqrt{2}}{7 (\sqrt{2} \sqrt{2})}$

Step 7.3

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

$\frac{3 \sqrt{13} \sqrt{2}}{7 ({\sqrt{2}}^{1} \sqrt{2})}$

Step 7.4

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

$\frac{3 \sqrt{13} \sqrt{2}}{7 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}$

Step 7.5

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

$\frac{3 \sqrt{13} \sqrt{2}}{7 {\sqrt{2}}^{1 + 1}}$

Step 7.6

Add $1$ and $1$ .

$\frac{3 \sqrt{13} \sqrt{2}}{7 {\sqrt{2}}^{2}}$

Step 7.7

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

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Step 7.7.1

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

$\frac{3 \sqrt{13} \sqrt{2}}{7 {(2^{\frac{1}{2}})}^{2}}$

Step 7.7.2

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

$\frac{3 \sqrt{13} \sqrt{2}}{7 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 7.7.3

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

$\frac{3 \sqrt{13} \sqrt{2}}{7 \cdot 2^{\frac{2}{2}}}$

Step 7.7.4

Cancel the common factor of $2$ .

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Step 7.7.4.1

Cancel the common factor.

$\frac{3 \sqrt{13} \sqrt{2}}{7 \cdot 2^{\frac{2}{2}}}$

Step 7.7.4.2

Rewrite the expression.

$\frac{3 \sqrt{13} \sqrt{2}}{7 \cdot 2^{1}}$

Step 7.7.5

Evaluate the exponent.

$\frac{3 \sqrt{13} \sqrt{2}}{7 \cdot 2}$

Step 8

Simplify the numerator.

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Step 8.1

Combine using the product rule for radicals.

$\frac{3 \sqrt{2 \cdot 13}}{7 \cdot 2}$

Step 8.2

Multiply $2$ by $13$ .

$\frac{3 \sqrt{26}}{7 \cdot 2}$

Step 9

Multiply $7$ by $2$ .

$\frac{3 \sqrt{26}}{14}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{3 \sqrt{26}}{14}$

Decimal Form:

$1.09264703 \dots$