Algebra Examples

$\sqrt{\frac{5}{6}} = \frac{\sqrt{5}}{\sqrt{6}}$

Step 1

Subtract $\frac{\sqrt{5}}{\sqrt{6}}$ from both sides of the equation.

$\sqrt{\frac{5}{6}} - \frac{\sqrt{5}}{\sqrt{6}} = 0$

Step 2

Simplify $\sqrt{\frac{5}{6}} - \frac{\sqrt{5}}{\sqrt{6}}$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite $\sqrt{\frac{5}{6}}$ as $\frac{\sqrt{5}}{\sqrt{6}}$ .

$\frac{\sqrt{5}}{\sqrt{6}} - \frac{\sqrt{5}}{\sqrt{6}} = 0$

Step 2.1.2

Multiply $\frac{\sqrt{5}}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\frac{\sqrt{5}}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}} - \frac{\sqrt{5}}{\sqrt{6}} = 0$

Step 2.1.3

Combine and simplify the denominator.

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Step 2.1.3.1

Multiply $\frac{\sqrt{5}}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\frac{\sqrt{5} \sqrt{6}}{\sqrt{6} \sqrt{6}} - \frac{\sqrt{5}}{\sqrt{6}} = 0$

Step 2.1.3.2

Raise $\sqrt{6}$ to the power of $1$ .

$\frac{\sqrt{5} \sqrt{6}}{{\sqrt{6}}^{1} \sqrt{6}} - \frac{\sqrt{5}}{\sqrt{6}} = 0$

Step 2.1.3.3

Raise $\sqrt{6}$ to the power of $1$ .

$\frac{\sqrt{5} \sqrt{6}}{{\sqrt{6}}^{1} {\sqrt{6}}^{1}} - \frac{\sqrt{5}}{\sqrt{6}} = 0$

Step 2.1.3.4

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

$\frac{\sqrt{5} \sqrt{6}}{{\sqrt{6}}^{1 + 1}} - \frac{\sqrt{5}}{\sqrt{6}} = 0$

Step 2.1.3.5

Add $1$ and $1$ .

$\frac{\sqrt{5} \sqrt{6}}{{\sqrt{6}}^{2}} - \frac{\sqrt{5}}{\sqrt{6}} = 0$

Step 2.1.3.6

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 2.1.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\frac{\sqrt{5} \sqrt{6}}{{(6^{\frac{1}{2}})}^{2}} - \frac{\sqrt{5}}{\sqrt{6}} = 0$

Step 2.1.3.6.2

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

$\frac{\sqrt{5} \sqrt{6}}{6^{\frac{1}{2} \cdot 2}} - \frac{\sqrt{5}}{\sqrt{6}} = 0$

Step 2.1.3.6.3

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

$\frac{\sqrt{5} \sqrt{6}}{6^{\frac{2}{2}}} - \frac{\sqrt{5}}{\sqrt{6}} = 0$

Step 2.1.3.6.4

Cancel the common factor of $2$ .

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Step 2.1.3.6.4.1

Cancel the common factor.

$\frac{\sqrt{5} \sqrt{6}}{6^{\frac{2}{2}}} - \frac{\sqrt{5}}{\sqrt{6}} = 0$

Step 2.1.3.6.4.2

Rewrite the expression.

$\frac{\sqrt{5} \sqrt{6}}{6^{1}} - \frac{\sqrt{5}}{\sqrt{6}} = 0$

Step 2.1.3.6.5

Evaluate the exponent.

$\frac{\sqrt{5} \sqrt{6}}{6} - \frac{\sqrt{5}}{\sqrt{6}} = 0$

Step 2.1.4

Simplify the numerator.

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Step 2.1.4.1

Combine using the product rule for radicals.

$\frac{\sqrt{5 \cdot 6}}{6} - \frac{\sqrt{5}}{\sqrt{6}} = 0$

Step 2.1.4.2

Multiply $5$ by $6$ .

$\frac{\sqrt{30}}{6} - \frac{\sqrt{5}}{\sqrt{6}} = 0$

Step 2.1.5

Multiply $\frac{\sqrt{5}}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\frac{\sqrt{30}}{6} - (\frac{\sqrt{5}}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}) = 0$

Step 2.1.6

Combine and simplify the denominator.

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Step 2.1.6.1

Multiply $\frac{\sqrt{5}}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\frac{\sqrt{30}}{6} - \frac{\sqrt{5} \sqrt{6}}{\sqrt{6} \sqrt{6}} = 0$

Step 2.1.6.2

Raise $\sqrt{6}$ to the power of $1$ .

$\frac{\sqrt{30}}{6} - \frac{\sqrt{5} \sqrt{6}}{{\sqrt{6}}^{1} \sqrt{6}} = 0$

Step 2.1.6.3

Raise $\sqrt{6}$ to the power of $1$ .

$\frac{\sqrt{30}}{6} - \frac{\sqrt{5} \sqrt{6}}{{\sqrt{6}}^{1} {\sqrt{6}}^{1}} = 0$

Step 2.1.6.4

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

$\frac{\sqrt{30}}{6} - \frac{\sqrt{5} \sqrt{6}}{{\sqrt{6}}^{1 + 1}} = 0$

Step 2.1.6.5

Add $1$ and $1$ .

$\frac{\sqrt{30}}{6} - \frac{\sqrt{5} \sqrt{6}}{{\sqrt{6}}^{2}} = 0$

Step 2.1.6.6

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 2.1.6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\frac{\sqrt{30}}{6} - \frac{\sqrt{5} \sqrt{6}}{{(6^{\frac{1}{2}})}^{2}} = 0$

Step 2.1.6.6.2

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

$\frac{\sqrt{30}}{6} - \frac{\sqrt{5} \sqrt{6}}{6^{\frac{1}{2} \cdot 2}} = 0$

Step 2.1.6.6.3

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

$\frac{\sqrt{30}}{6} - \frac{\sqrt{5} \sqrt{6}}{6^{\frac{2}{2}}} = 0$

Step 2.1.6.6.4

Cancel the common factor of $2$ .

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Step 2.1.6.6.4.1

Cancel the common factor.

$\frac{\sqrt{30}}{6} - \frac{\sqrt{5} \sqrt{6}}{6^{\frac{2}{2}}} = 0$

Step 2.1.6.6.4.2

Rewrite the expression.

$\frac{\sqrt{30}}{6} - \frac{\sqrt{5} \sqrt{6}}{6^{1}} = 0$

Step 2.1.6.6.5

Evaluate the exponent.

$\frac{\sqrt{30}}{6} - \frac{\sqrt{5} \sqrt{6}}{6} = 0$

Step 2.1.7

Simplify the numerator.

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Step 2.1.7.1

Combine using the product rule for radicals.

$\frac{\sqrt{30}}{6} - \frac{\sqrt{5 \cdot 6}}{6} = 0$

Step 2.1.7.2

Multiply $5$ by $6$ .

$\frac{\sqrt{30}}{6} - \frac{\sqrt{30}}{6} = 0$

Step 2.2

Combine the numerators over the common denominator.

$\frac{\sqrt{30} - \sqrt{30}}{6} = 0$

Step 2.3

Subtract $\sqrt{30}$ from $\sqrt{30}$ .

$\frac{0}{6} = 0$

Step 2.4

Divide $0$ by $6$ .

$0 = 0$

Step 3

Since $0 = 0$ , the equation will always be true.

Always true