Algebra Examples

$\sqrt{2} \cdot \sqrt[3]{3} = \sqrt[6]{x}$

Step 1

Rewrite the equation as $\sqrt[6]{x} = \sqrt{2} \cdot \sqrt[3]{3}$ .

$\sqrt[6]{x} = \sqrt{2} \cdot \sqrt[3]{3}$

Step 2

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $6$ .

${\sqrt[6]{x}}^{6} = {(\sqrt{2} \cdot \sqrt[3]{3})}^{6}$

Step 3

Simplify each side of the equation.

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Step 3.1

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

${(x^{\frac{1}{6}})}^{6} = {(\sqrt{2} \cdot \sqrt[3]{3})}^{6}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${(x^{\frac{1}{6}})}^{6}$ .

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Step 3.2.1.1

Multiply the exponents in ${(x^{\frac{1}{6}})}^{6}$ .

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Step 3.2.1.1.1

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

$x^{\frac{1}{6} \cdot 6} = {(\sqrt{2} \cdot \sqrt[3]{3})}^{6}$

Step 3.2.1.1.2

Cancel the common factor of $6$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{6} \cdot 6} = {(\sqrt{2} \cdot \sqrt[3]{3})}^{6}$

Step 3.2.1.1.2.2

Rewrite the expression.

$x^{1} = {(\sqrt{2} \cdot \sqrt[3]{3})}^{6}$

Step 3.2.1.2

Simplify.

$x = {(\sqrt{2} \cdot \sqrt[3]{3})}^{6}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(\sqrt{2} \cdot \sqrt[3]{3})}^{6}$ .

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Step 3.3.1.1

Multiply $\sqrt{2} \sqrt[3]{3}$ .

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Step 3.3.1.1.1

Rewrite the expression using the least common index of $6$ .

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Step 3.3.1.1.1.1

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

$x = {(2^{\frac{1}{2}} \sqrt[3]{3})}^{6}$

Step 3.3.1.1.1.2

Rewrite $2^{\frac{1}{2}}$ as $2^{\frac{3}{6}}$ .

$x = {(2^{\frac{3}{6}} \sqrt[3]{3})}^{6}$

Step 3.3.1.1.1.3

Rewrite $2^{\frac{3}{6}}$ as $\sqrt[6]{2^{3}}$ .

$x = {(\sqrt[6]{2^{3}} \sqrt[3]{3})}^{6}$

Step 3.3.1.1.1.4

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

$x = {(\sqrt[6]{2^{3}} \cdot 3^{\frac{1}{3}})}^{6}$

Step 3.3.1.1.1.5

Rewrite $3^{\frac{1}{3}}$ as $3^{\frac{2}{6}}$ .

$x = {(\sqrt[6]{2^{3}} \cdot 3^{\frac{2}{6}})}^{6}$

Step 3.3.1.1.1.6

Rewrite $3^{\frac{2}{6}}$ as $\sqrt[6]{3^{2}}$ .

$x = {(\sqrt[6]{2^{3}} \sqrt[6]{3^{2}})}^{6}$

Step 3.3.1.1.2

Combine using the product rule for radicals.

$x = {\sqrt[6]{2^{3} \cdot 3^{2}}}^{6}$

Step 3.3.1.1.3

Raise $2$ to the power of $3$ .

$x = {\sqrt[6]{8 \cdot 3^{2}}}^{6}$

Step 3.3.1.1.4

Raise $3$ to the power of $2$ .

$x = {\sqrt[6]{8 \cdot 9}}^{6}$

Step 3.3.1.1.5

Multiply $8$ by $9$ .

$x = {\sqrt[6]{72}}^{6}$

Step 3.3.1.2

Rewrite ${\sqrt[6]{72}}^{6}$ as $72$ .

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Step 3.3.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[6]{72}$ as $72^{\frac{1}{6}}$ .

$x = {(72^{\frac{1}{6}})}^{6}$

Step 3.3.1.2.2

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

$x = 72^{\frac{1}{6} \cdot 6}$

Step 3.3.1.2.3

Combine $\frac{1}{6}$ and $6$ .

$x = 72^{\frac{6}{6}}$

Step 3.3.1.2.4

Cancel the common factor of $6$ .

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Step 3.3.1.2.4.1

Cancel the common factor.

$x = 72^{\frac{6}{6}}$

Step 3.3.1.2.4.2

Rewrite the expression.

$x = 72^{1}$

Step 3.3.1.2.5

Evaluate the exponent.

$x = 72$