Simplify the Radical Expression fourth root of 3^3* square root of 3^5

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

Step 1

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

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

Step 2

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

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

Step 3

Rewrite $243$ as $9^{2} \cdot 3$ .

Factor $81$ out of $243$ .

$\sqrt[4]{27} \cdot \sqrt{81 (3)}$

Rewrite $81$ as $9^{2}$ .

$\sqrt[4]{27} \cdot \sqrt{9^{2} \cdot 3}$

Step 4

Pull terms out from under the radical.

$\sqrt[4]{27} \cdot (| 9 | \sqrt{3})$

Step 5

The absolute value is the distance between a number and zero. The distance between $0$ and $9$ is $9$ .

$\sqrt[4]{27} \cdot (9 \sqrt{3})$

Step 6

Multiply $\sqrt[4]{27} (9 \sqrt{3})$ .

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

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$9 (\sqrt[4]{27} \cdot 3^{\frac{1}{2}})$

Rewrite $3^{\frac{1}{2}}$ as $3^{\frac{2}{4}}$ .

$9 (\sqrt[4]{27} \cdot 3^{\frac{2}{4}})$

Rewrite $3^{\frac{2}{4}}$ as $\sqrt[4]{3^{2}}$ .

$9 (\sqrt[4]{27} \sqrt[4]{3^{2}})$

Combine using the product rule for radicals.

$9 \sqrt[4]{27 \cdot 3^{2}}$

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

$9 \sqrt[4]{3^{3} \cdot 3^{2}}$

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

$9 \sqrt[4]{3^{3 + 2}}$

Add $3$ and $2$ .

$9 \sqrt[4]{3^{5}}$

Step 7

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

$9 \sqrt[4]{243}$

Step 8

Rewrite $243$ as $3^{4} \cdot 3$ .

Factor $81$ out of $243$ .

$9 \sqrt[4]{81 (3)}$

Rewrite $81$ as $3^{4}$ .

$9 \sqrt[4]{3^{4} \cdot 3}$

Step 9

Pull terms out from under the radical.

$9 (| 3 | \sqrt[4]{3})$

Step 10

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

$9 (3 \sqrt[4]{3})$

Step 11

Multiply $3$ by $9$ .

$27 \sqrt[4]{3}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$27 \sqrt[4]{3}$

Decimal Form:

$35.53399834 \dots$

Simplify the Radical Expression fourth root of 3^3* square root of 3^5

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