Algebra Examples

$\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}$

Step 1

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $n$ .

${\sqrt[n]{\frac{a}{b}}}^{n} = {(\frac{\sqrt[n]{a}}{\sqrt[n]{b}})}^{n}$

Step 2

Simplify each side of the equation.

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Step 2.1

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

${({(\frac{a}{b})}^{\frac{1}{n}})}^{n} = {(\frac{\sqrt[n]{a}}{\sqrt[n]{b}})}^{n}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(\frac{a}{b})}^{\frac{1}{n}})}^{n}$ .

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Step 2.2.1.1

Apply basic rules of exponents.

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Step 2.2.1.1.1

Apply the product rule to $\frac{a}{b}$ .

${(\frac{a^{\frac{1}{n}}}{b^{\frac{1}{n}}})}^{n} = {(\frac{\sqrt[n]{a}}{\sqrt[n]{b}})}^{n}$

Step 2.2.1.1.2

Apply the product rule to $\frac{a^{\frac{1}{n}}}{b^{\frac{1}{n}}}$ .

$\frac{{(a^{\frac{1}{n}})}^{n}}{{(b^{\frac{1}{n}})}^{n}} = {(\frac{\sqrt[n]{a}}{\sqrt[n]{b}})}^{n}$

Step 2.2.1.2

Simplify the numerator.

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Step 2.2.1.2.1

Multiply the exponents in ${(a^{\frac{1}{n}})}^{n}$ .

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Step 2.2.1.2.1.1

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

$\frac{a^{\frac{1}{n} n}}{{(b^{\frac{1}{n}})}^{n}} = {(\frac{\sqrt[n]{a}}{\sqrt[n]{b}})}^{n}$

Step 2.2.1.2.1.2

Cancel the common factor of $n$ .

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Step 2.2.1.2.1.2.1

Cancel the common factor.

$\frac{a^{\frac{1}{n} n}}{{(b^{\frac{1}{n}})}^{n}} = {(\frac{\sqrt[n]{a}}{\sqrt[n]{b}})}^{n}$

Step 2.2.1.2.1.2.2

Rewrite the expression.

$\frac{a^{1}}{{(b^{\frac{1}{n}})}^{n}} = {(\frac{\sqrt[n]{a}}{\sqrt[n]{b}})}^{n}$

Step 2.2.1.2.2

Simplify.

$\frac{a}{{(b^{\frac{1}{n}})}^{n}} = {(\frac{\sqrt[n]{a}}{\sqrt[n]{b}})}^{n}$

Step 2.2.1.3

Simplify the denominator.

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Step 2.2.1.3.1

Multiply the exponents in ${(b^{\frac{1}{n}})}^{n}$ .

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Step 2.2.1.3.1.1

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

$\frac{a}{b^{\frac{1}{n} n}} = {(\frac{\sqrt[n]{a}}{\sqrt[n]{b}})}^{n}$

Step 2.2.1.3.1.2

Cancel the common factor of $n$ .

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Step 2.2.1.3.1.2.1

Cancel the common factor.

$\frac{a}{b^{\frac{1}{n} n}} = {(\frac{\sqrt[n]{a}}{\sqrt[n]{b}})}^{n}$

Step 2.2.1.3.1.2.2

Rewrite the expression.

$\frac{a}{b^{1}} = {(\frac{\sqrt[n]{a}}{\sqrt[n]{b}})}^{n}$

Step 2.2.1.3.2

Simplify.

$\frac{a}{b} = {(\frac{\sqrt[n]{a}}{\sqrt[n]{b}})}^{n}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${(\frac{\sqrt[n]{a}}{\sqrt[n]{b}})}^{n}$ .

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Step 2.3.1.1

Apply the product rule to $\frac{\sqrt[n]{a}}{\sqrt[n]{b}}$ .

$\frac{a}{b} = \frac{{\sqrt[n]{a}}^{n}}{{\sqrt[n]{b}}^{n}}$

Step 2.3.1.2

Rewrite ${\sqrt[n]{a}}^{n}$ as $a$ .

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Step 2.3.1.2.1

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

$\frac{a}{b} = \frac{{(a^{\frac{1}{n}})}^{n}}{{\sqrt[n]{b}}^{n}}$

Step 2.3.1.2.2

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

$\frac{a}{b} = \frac{a^{\frac{1}{n} n}}{{\sqrt[n]{b}}^{n}}$

Step 2.3.1.2.3

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

$\frac{a}{b} = \frac{a^{\frac{n}{n}}}{{\sqrt[n]{b}}^{n}}$

Step 2.3.1.2.4

Cancel the common factor of $n$ .

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Step 2.3.1.2.4.1

Cancel the common factor.

$\frac{a}{b} = \frac{a^{\frac{n}{n}}}{{\sqrt[n]{b}}^{n}}$

Step 2.3.1.2.4.2

Rewrite the expression.

$\frac{a}{b} = \frac{a^{1}}{{\sqrt[n]{b}}^{n}}$

Step 2.3.1.2.5

Simplify.

$\frac{a}{b} = \frac{a}{{\sqrt[n]{b}}^{n}}$

Step 2.3.1.3

Rewrite ${\sqrt[n]{b}}^{n}$ as $b$ .

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Step 2.3.1.3.1

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

$\frac{a}{b} = \frac{a}{{(b^{\frac{1}{n}})}^{n}}$

Step 2.3.1.3.2

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

$\frac{a}{b} = \frac{a}{b^{\frac{1}{n} n}}$

Step 2.3.1.3.3

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

$\frac{a}{b} = \frac{a}{b^{\frac{n}{n}}}$

Step 2.3.1.3.4

Cancel the common factor of $n$ .

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Step 2.3.1.3.4.1

Cancel the common factor.

$\frac{a}{b} = \frac{a}{b^{\frac{n}{n}}}$

Step 2.3.1.3.4.2

Rewrite the expression.

$\frac{a}{b} = \frac{a}{b^{1}}$

Step 2.3.1.3.5

Simplify.

$\frac{a}{b} = \frac{a}{b}$

Step 3

Solve for $n$ .

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Step 3.1

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$a = a$

Step 3.2

Move all terms containing $a$ to the left side of the equation.

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Step 3.2.1

Subtract $a$ from both sides of the equation.

$a - a = 0$

Step 3.2.2

Subtract $a$ from $a$ .

$0 = 0$

Step 3.3

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

Always true

Step 4

The result can be shown in multiple forms.

Always true

Interval Notation:

$(- \infty, \infty)$