Algebra Examples

$\sqrt[4]{3^{3}} \cdot \sqrt[6]{3^{5}}$

Step 1

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

$\sqrt[4]{27} \cdot \sqrt[6]{3^{5}}$

Step 2

Raise $3$ to the power of $5$ .

$\sqrt[4]{27} \cdot \sqrt[6]{243}$

Step 3

Multiply $\sqrt[4]{27} \sqrt[6]{243}$ .

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Step 3.1

Rewrite the expression using the least common index of $12$ .

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Step 3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{27}$ as $27^{\frac{1}{4}}$ .

$27^{\frac{1}{4}} \sqrt[6]{243}$

Step 3.1.2

Rewrite $27^{\frac{1}{4}}$ as $27^{\frac{3}{12}}$ .

$27^{\frac{3}{12}} \sqrt[6]{243}$

Step 3.1.3

Rewrite $27^{\frac{3}{12}}$ as $\sqrt[12]{27^{3}}$ .

$\sqrt[12]{27^{3}} \sqrt[6]{243}$

Step 3.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[6]{243}$ as $243^{\frac{1}{6}}$ .

$\sqrt[12]{27^{3}} \cdot 243^{\frac{1}{6}}$

Step 3.1.5

Rewrite $243^{\frac{1}{6}}$ as $243^{\frac{2}{12}}$ .

$\sqrt[12]{27^{3}} \cdot 243^{\frac{2}{12}}$

Step 3.1.6

Rewrite $243^{\frac{2}{12}}$ as $\sqrt[12]{243^{2}}$ .

$\sqrt[12]{27^{3}} \sqrt[12]{243^{2}}$

Step 3.2

Combine using the product rule for radicals.

$\sqrt[12]{27^{3} \cdot 243^{2}}$

Step 3.3

Rewrite $27$ as $3^{3}$ .

$\sqrt[12]{{(3^{3})}^{3} \cdot 243^{2}}$

Step 3.4

Multiply the exponents in ${(3^{3})}^{3}$ .

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Step 3.4.1

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

$\sqrt[12]{3^{3 \cdot 3} \cdot 243^{2}}$

Step 3.4.2

Multiply $3$ by $3$ .

$\sqrt[12]{3^{9} \cdot 243^{2}}$

Step 3.5

Rewrite $243$ as $3^{5}$ .

$\sqrt[12]{3^{9} \cdot {(3^{5})}^{2}}$

Step 3.6

Multiply the exponents in ${(3^{5})}^{2}$ .

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Step 3.6.1

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

$\sqrt[12]{3^{9} \cdot 3^{5 \cdot 2}}$

Step 3.6.2

Multiply $5$ by $2$ .

$\sqrt[12]{3^{9} \cdot 3^{10}}$

Step 3.7

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

$\sqrt[12]{3^{9 + 10}}$

Step 3.8

Add $9$ and $10$ .

$\sqrt[12]{3^{19}}$

Step 4

Raise $3$ to the power of $19$ .

$\sqrt[12]{1162261467}$

Step 5

Rewrite $1162261467$ as $3^{12} \cdot 2187$ .

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Step 5.1

Factor $531441$ out of $1162261467$ .

$\sqrt[12]{531441 (2187)}$

Step 5.2

Rewrite $531441$ as $3^{12}$ .

$\sqrt[12]{3^{12} \cdot 2187}$

Step 6

Pull terms out from under the radical.

$| 3 | \sqrt[12]{2187}$

Step 7

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$3 \sqrt[12]{2187}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$3 \sqrt[12]{2187}$

Decimal Form:

$5.69432153 \dots$