Simplify sixth root of 3^5* fourth root of 3^3

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

Step 1

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

$\sqrt[6]{243} \cdot \sqrt[4]{3^{3}}$

Step 2

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

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

Step 3

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

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

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

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

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

$243^{\frac{2}{12}} \sqrt[4]{27}$

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

$\sqrt[12]{243^{2}} \sqrt[4]{27}$

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

$\sqrt[12]{243^{2}} \cdot 27^{\frac{1}{4}}$

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

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

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

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

Combine using the product rule for radicals.

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

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

$\sqrt[12]{{(3^{5})}^{2} \cdot 27^{3}}$

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

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

$\sqrt[12]{3^{5 \cdot 2} \cdot 27^{3}}$

Multiply $5$ by $2$ .

$\sqrt[12]{3^{10} \cdot 27^{3}}$

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

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

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

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

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

Multiply $3$ by $3$ .

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

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

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

Add $10$ and $9$ .

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

Step 4

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

$\sqrt[12]{1162261467}$

Step 5

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

Factor $531441$ out of $1162261467$ .

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

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 result can be shown in multiple forms.

Exact Form:

$3 \sqrt[12]{2187}$

Decimal Form:

$5.69432153 \dots$

Simplify sixth root of 3^5* fourth root of 3^3

Related Questions

Support LCMGCF.com by disabling your adblocker.