Pre-Algebra Examples

$\frac{\sqrt{126}}{\sqrt{35}}$

Step 1

Combine $\sqrt{126}$ and $\sqrt{35}$ into a single radical.

$\sqrt{\frac{126}{35}}$

Step 2

Cancel the common factor of $126$ and $35$ .

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Step 2.1

Factor $7$ out of $126$ .

$\sqrt{\frac{7 (18)}{35}}$

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor $7$ out of $35$ .

$\sqrt{\frac{7 \cdot 18}{7 \cdot 5}}$

Step 2.2.2

Cancel the common factor.

$\sqrt{\frac{7 \cdot 18}{7 \cdot 5}}$

Step 2.2.3

Rewrite the expression.

$\sqrt{\frac{18}{5}}$

Step 3

Rewrite $\sqrt{\frac{18}{5}}$ as $\frac{\sqrt{18}}{\sqrt{5}}$ .

$\frac{\sqrt{18}}{\sqrt{5}}$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 4.1.1

Factor $9$ out of $18$ .

$\frac{\sqrt{9 (2)}}{\sqrt{5}}$

Step 4.1.2

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

$\frac{\sqrt{3^{2} \cdot 2}}{\sqrt{5}}$

Step 4.2

Pull terms out from under the radical.

$\frac{3 \sqrt{2}}{\sqrt{5}}$

Step 5

Multiply $\frac{3 \sqrt{2}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{3 \sqrt{2}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 6

Combine and simplify the denominator.

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Step 6.1

Multiply $\frac{3 \sqrt{2}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{3 \sqrt{2} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 6.2

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{3 \sqrt{2} \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Step 6.3

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{3 \sqrt{2} \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

Step 6.4

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

$\frac{3 \sqrt{2} \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 6.5

Add $1$ and $1$ .

$\frac{3 \sqrt{2} \sqrt{5}}{{\sqrt{5}}^{2}}$

Step 6.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{3 \sqrt{2} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 6.6.2

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

$\frac{3 \sqrt{2} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 6.6.3

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

$\frac{3 \sqrt{2} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 6.6.4

Cancel the common factor of $2$ .

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Step 6.6.4.1

Cancel the common factor.

$\frac{3 \sqrt{2} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 6.6.4.2

Rewrite the expression.

$\frac{3 \sqrt{2} \sqrt{5}}{5^{1}}$

Step 6.6.5

Evaluate the exponent.

$\frac{3 \sqrt{2} \sqrt{5}}{5}$

Step 7

Simplify the numerator.

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Step 7.1

Combine using the product rule for radicals.

$\frac{3 \sqrt{5 \cdot 2}}{5}$

Step 7.2

Multiply $5$ by $2$ .

$\frac{3 \sqrt{10}}{5}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{3 \sqrt{10}}{5}$

Decimal Form:

$1.89736659 \dots$