Algebra Examples

$\sqrt[6]{a} \cdot \sqrt[3]{a} \cdot \sqrt[2]{a}$

Step 1

Multiply $\sqrt[6]{a} \sqrt[3]{a}$ .

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Step 1.1

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

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Step 1.1.1

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

$\sqrt[6]{a} a^{\frac{1}{3}} \cdot \sqrt{a}$

Step 1.1.2

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

$\sqrt[6]{a} a^{\frac{2}{6}} \cdot \sqrt{a}$

Step 1.1.3

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

$\sqrt[6]{a} \sqrt[6]{a^{2}} \cdot \sqrt{a}$

Step 1.2

Combine using the product rule for radicals.

$\sqrt[6]{a \cdot a^{2}} \cdot \sqrt{a}$

Step 1.3

Multiply $a$ by $a^{2}$ by adding the exponents.

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Step 1.3.1

Multiply $a$ by $a^{2}$ .

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Step 1.3.1.1

Raise $a$ to the power of $1$ .

$\sqrt[6]{a^{1} a^{2}} \cdot \sqrt{a}$

Step 1.3.1.2

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

$\sqrt[6]{a^{1 + 2}} \cdot \sqrt{a}$

Step 1.3.2

Add $1$ and $2$ .

$\sqrt[6]{a^{3}} \cdot \sqrt{a}$

Step 2

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

$\sqrt{\sqrt[3]{a^{3}}} \cdot \sqrt{a}$

Step 3

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

$\sqrt{a} \cdot \sqrt{a}$

Step 4

Multiply $\sqrt{a} \sqrt{a}$ .

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Step 4.1

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

${\sqrt{a}}^{1} \sqrt{a}$

Step 4.2

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

${\sqrt{a}}^{1} {\sqrt{a}}^{1}$

Step 4.3

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

${\sqrt{a}}^{1 + 1}$

Step 4.4

Add $1$ and $1$ .

${\sqrt{a}}^{2}$

Step 5

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

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Step 5.1

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

${(a^{\frac{1}{2}})}^{2}$

Step 5.2

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

$a^{\frac{1}{2} \cdot 2}$

Step 5.3

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

$a^{\frac{2}{2}}$

Step 5.4

Cancel the common factor of $2$ .

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Step 5.4.1

Cancel the common factor.

$a^{\frac{2}{2}}$

Step 5.4.2

Rewrite the expression.

$a^{1}$

Step 5.5

Simplify.

$a$