Algebra Examples

$\sqrt[4]{\frac{7^{2}}{m}} = \frac{\sqrt[4]{7^{2}}}{\sqrt[4]{m}}$

Step 1

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $4$ .

${\sqrt[4]{\frac{7^{2}}{m}}}^{4} = {(\frac{\sqrt[4]{7^{2}}}{\sqrt[4]{m}})}^{4}$

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 $\sqrt[4]{\frac{7^{2}}{m}}$ as ${(\frac{7^{2}}{m})}^{\frac{1}{4}}$ .

${({(\frac{7^{2}}{m})}^{\frac{1}{4}})}^{4} = {(\frac{\sqrt[4]{7^{2}}}{\sqrt[4]{m}})}^{4}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(\frac{7^{2}}{m})}^{\frac{1}{4}})}^{4}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(\frac{7^{2}}{m})}^{\frac{1}{4}})}^{4}$ .

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Step 2.2.1.1.1

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

${(\frac{7^{2}}{m})}^{\frac{1}{4} \cdot 4} = {(\frac{\sqrt[4]{7^{2}}}{\sqrt[4]{m}})}^{4}$

Step 2.2.1.1.2

Cancel the common factor of $4$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

${(\frac{7^{2}}{m})}^{\frac{1}{4} \cdot 4} = {(\frac{\sqrt[4]{7^{2}}}{\sqrt[4]{m}})}^{4}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(\frac{7^{2}}{m})}^{1} = {(\frac{\sqrt[4]{7^{2}}}{\sqrt[4]{m}})}^{4}$

Step 2.2.1.2

Raise $7$ to the power of $2$ .

${(\frac{49}{m})}^{1} = {(\frac{\sqrt[4]{7^{2}}}{\sqrt[4]{m}})}^{4}$

Step 2.2.1.3

Simplify.

$\frac{49}{m} = {(\frac{\sqrt[4]{7^{2}}}{\sqrt[4]{m}})}^{4}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${(\frac{\sqrt[4]{7^{2}}}{\sqrt[4]{m}})}^{4}$ .

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Step 2.3.1.1

Simplify the numerator.

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Step 2.3.1.1.1

Raise $7$ to the power of $2$ .

$\frac{49}{m} = {(\frac{\sqrt[4]{49}}{\sqrt[4]{m}})}^{4}$

Step 2.3.1.1.2

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

$\frac{49}{m} = {(\frac{\sqrt[4]{7^{2}}}{\sqrt[4]{m}})}^{4}$

Step 2.3.1.1.3

Rewrite $\sqrt[4]{7^{2}}$ as $\sqrt{\sqrt{7^{2}}}$ .

$\frac{49}{m} = {(\frac{\sqrt{\sqrt{7^{2}}}}{\sqrt[4]{m}})}^{4}$

Step 2.3.1.1.4

Pull terms out from under the radical, assuming positive real numbers.

$\frac{49}{m} = {(\frac{\sqrt{7}}{\sqrt[4]{m}})}^{4}$

Step 2.3.1.2

Multiply $\frac{\sqrt{7}}{\sqrt[4]{m}}$ by $\frac{{\sqrt[4]{m}}^{3}}{{\sqrt[4]{m}}^{3}}$ .

$\frac{49}{m} = {(\frac{\sqrt{7}}{\sqrt[4]{m}} \cdot \frac{{\sqrt[4]{m}}^{3}}{{\sqrt[4]{m}}^{3}})}^{4}$

Step 2.3.1.3

Combine and simplify the denominator.

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Step 2.3.1.3.1

Multiply $\frac{\sqrt{7}}{\sqrt[4]{m}}$ by $\frac{{\sqrt[4]{m}}^{3}}{{\sqrt[4]{m}}^{3}}$ .

$\frac{49}{m} = {(\frac{\sqrt{7} {\sqrt[4]{m}}^{3}}{\sqrt[4]{m} {\sqrt[4]{m}}^{3}})}^{4}$

Step 2.3.1.3.2

Raise $\sqrt[4]{m}$ to the power of $1$ .

$\frac{49}{m} = {(\frac{\sqrt{7} {\sqrt[4]{m}}^{3}}{{\sqrt[4]{m}}^{1} {\sqrt[4]{m}}^{3}})}^{4}$

Step 2.3.1.3.3

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

$\frac{49}{m} = {(\frac{\sqrt{7} {\sqrt[4]{m}}^{3}}{{\sqrt[4]{m}}^{1 + 3}})}^{4}$

Step 2.3.1.3.4

Add $1$ and $3$ .

$\frac{49}{m} = {(\frac{\sqrt{7} {\sqrt[4]{m}}^{3}}{{\sqrt[4]{m}}^{4}})}^{4}$

Step 2.3.1.3.5

Rewrite ${\sqrt[4]{m}}^{4}$ as $m$ .

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Step 2.3.1.3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{m}$ as $m^{\frac{1}{4}}$ .

$\frac{49}{m} = {(\frac{\sqrt{7} {\sqrt[4]{m}}^{3}}{{(m^{\frac{1}{4}})}^{4}})}^{4}$

Step 2.3.1.3.5.2

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

$\frac{49}{m} = {(\frac{\sqrt{7} {\sqrt[4]{m}}^{3}}{m^{\frac{1}{4} \cdot 4}})}^{4}$

Step 2.3.1.3.5.3

Combine $\frac{1}{4}$ and $4$ .

$\frac{49}{m} = {(\frac{\sqrt{7} {\sqrt[4]{m}}^{3}}{m^{\frac{4}{4}}})}^{4}$

Step 2.3.1.3.5.4

Cancel the common factor of $4$ .

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Step 2.3.1.3.5.4.1

Cancel the common factor.

$\frac{49}{m} = {(\frac{\sqrt{7} {\sqrt[4]{m}}^{3}}{m^{\frac{4}{4}}})}^{4}$

Step 2.3.1.3.5.4.2

Rewrite the expression.

$\frac{49}{m} = {(\frac{\sqrt{7} {\sqrt[4]{m}}^{3}}{m^{1}})}^{4}$

Step 2.3.1.3.5.5

Simplify.

$\frac{49}{m} = {(\frac{\sqrt{7} {\sqrt[4]{m}}^{3}}{m})}^{4}$

Step 2.3.1.4

Rewrite ${\sqrt[4]{m}}^{3}$ as $\sqrt[4]{m^{3}}$ .

$\frac{49}{m} = {(\frac{\sqrt{7} \sqrt[4]{m^{3}}}{m})}^{4}$

Step 2.3.1.5

Simplify the numerator.

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Step 2.3.1.5.1

Rewrite the expression using the least common index of $4$ .

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Step 2.3.1.5.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\frac{49}{m} = {(\frac{7^{\frac{1}{2}} \sqrt[4]{m^{3}}}{m})}^{4}$

Step 2.3.1.5.1.2

Rewrite $7^{\frac{1}{2}}$ as $7^{\frac{2}{4}}$ .

$\frac{49}{m} = {(\frac{7^{\frac{2}{4}} \sqrt[4]{m^{3}}}{m})}^{4}$

Step 2.3.1.5.1.3

Rewrite $7^{\frac{2}{4}}$ as $\sqrt[4]{7^{2}}$ .

$\frac{49}{m} = {(\frac{\sqrt[4]{7^{2}} \sqrt[4]{m^{3}}}{m})}^{4}$

Step 2.3.1.5.2

Combine using the product rule for radicals.

$\frac{49}{m} = {(\frac{\sqrt[4]{7^{2} m^{3}}}{m})}^{4}$

Step 2.3.1.5.3

Raise $7$ to the power of $2$ .

$\frac{49}{m} = {(\frac{\sqrt[4]{49 m^{3}}}{m})}^{4}$

Step 2.3.1.6

Simplify terms.

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Step 2.3.1.6.1

Apply the product rule to $\frac{\sqrt[4]{49 m^{3}}}{m}$ .

$\frac{49}{m} = \frac{{\sqrt[4]{49 m^{3}}}^{4}}{m^{4}}$

Step 2.3.1.6.2

Rewrite ${\sqrt[4]{49 m^{3}}}^{4}$ as $49 m^{3}$ .

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Step 2.3.1.6.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{49 m^{3}}$ as ${(49 m^{3})}^{\frac{1}{4}}$ .

$\frac{49}{m} = \frac{{({(49 m^{3})}^{\frac{1}{4}})}^{4}}{m^{4}}$

Step 2.3.1.6.2.2

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

$\frac{49}{m} = \frac{{(49 m^{3})}^{\frac{1}{4} \cdot 4}}{m^{4}}$

Step 2.3.1.6.2.3

Combine $\frac{1}{4}$ and $4$ .

$\frac{49}{m} = \frac{{(49 m^{3})}^{\frac{4}{4}}}{m^{4}}$

Step 2.3.1.6.2.4

Cancel the common factor of $4$ .

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Step 2.3.1.6.2.4.1

Cancel the common factor.

$\frac{49}{m} = \frac{{(49 m^{3})}^{\frac{4}{4}}}{m^{4}}$

Step 2.3.1.6.2.4.2

Rewrite the expression.

$\frac{49}{m} = \frac{{(49 m^{3})}^{1}}{m^{4}}$

Step 2.3.1.6.2.5

Simplify.

$\frac{49}{m} = \frac{49 m^{3}}{m^{4}}$

Step 2.3.1.6.3

Cancel the common factor of $m^{3}$ and $m^{4}$ .

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Step 2.3.1.6.3.1

Factor $m^{3}$ out of $49 m^{3}$ .

$\frac{49}{m} = \frac{m^{3} \cdot 49}{m^{4}}$

Step 2.3.1.6.3.2

Cancel the common factors.

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Step 2.3.1.6.3.2.1

Factor $m^{3}$ out of $m^{4}$ .

$\frac{49}{m} = \frac{m^{3} \cdot 49}{m^{3} m}$

Step 2.3.1.6.3.2.2

Cancel the common factor.

$\frac{49}{m} = \frac{m^{3} \cdot 49}{m^{3} m}$

Step 2.3.1.6.3.2.3

Rewrite the expression.

$\frac{49}{m} = \frac{49}{m}$

Step 3

Solve for $m$ .

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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.

$49 = 49$

Step 3.2

Since $49 = 49$ , the equation will always be true for any value of $m$ .

All real numbers

Step 4

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$