Evaluate cube root of 16/9

$\sqrt[3]{\frac{16}{9}}$

Step 1

Rewrite $\sqrt[3]{\frac{16}{9}}$ as $\frac{\sqrt[3]{16}}{\sqrt[3]{9}}$ .

$\frac{\sqrt[3]{16}}{\sqrt[3]{9}}$

Step 2

Simplify the numerator.

Rewrite $16$ as $2^{3} \cdot 2$ .

Factor $8$ out of $16$ .

$\frac{\sqrt[3]{8 (2)}}{\sqrt[3]{9}}$

Rewrite $8$ as $2^{3}$ .

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

Pull terms out from under the radical.

$\frac{2 \sqrt[3]{2}}{\sqrt[3]{9}}$

Step 3

Multiply $\frac{2 \sqrt[3]{2}}{\sqrt[3]{9}}$ by $\frac{{\sqrt[3]{9}}^{2}}{{\sqrt[3]{9}}^{2}}$ .

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

Step 4

Combine and simplify the denominator.

Multiply $\frac{2 \sqrt[3]{2}}{\sqrt[3]{9}}$ by $\frac{{\sqrt[3]{9}}^{2}}{{\sqrt[3]{9}}^{2}}$ .

$\frac{2 \sqrt[3]{2} {\sqrt[3]{9}}^{2}}{\sqrt[3]{9} {\sqrt[3]{9}}^{2}}$

Raise $\sqrt[3]{9}$ to the power of $1$ .

$\frac{2 \sqrt[3]{2} {\sqrt[3]{9}}^{2}}{{\sqrt[3]{9}}^{1} {\sqrt[3]{9}}^{2}}$

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

$\frac{2 \sqrt[3]{2} {\sqrt[3]{9}}^{2}}{{\sqrt[3]{9}}^{1 + 2}}$

Add $1$ and $2$ .

$\frac{2 \sqrt[3]{2} {\sqrt[3]{9}}^{2}}{{\sqrt[3]{9}}^{3}}$

Rewrite ${\sqrt[3]{9}}^{3}$ as $9$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{9}$ as $9^{\frac{1}{3}}$ .

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

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

$\frac{2 \sqrt[3]{2} {\sqrt[3]{9}}^{2}}{9^{\frac{1}{3} \cdot 3}}$

Combine $\frac{1}{3}$ and $3$ .

$\frac{2 \sqrt[3]{2} {\sqrt[3]{9}}^{2}}{9^{\frac{3}{3}}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{2 \sqrt[3]{2} {\sqrt[3]{9}}^{2}}{9^{\frac{3}{3}}}$

Rewrite the expression.

$\frac{2 \sqrt[3]{2} {\sqrt[3]{9}}^{2}}{9^{1}}$

Evaluate the exponent.

$\frac{2 \sqrt[3]{2} {\sqrt[3]{9}}^{2}}{9}$

Step 5

Simplify the numerator.

Rewrite ${\sqrt[3]{9}}^{2}$ as $\sqrt[3]{9^{2}}$ .

$\frac{2 \sqrt[3]{2} \sqrt[3]{9^{2}}}{9}$

Raise $9$ to the power of $2$ .

$\frac{2 \sqrt[3]{2} \sqrt[3]{81}}{9}$

Rewrite $81$ as $3^{3} \cdot 3$ .

Factor $27$ out of $81$ .

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

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

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

Pull terms out from under the radical.

$\frac{2 \sqrt[3]{2} \cdot 3 \sqrt[3]{3}}{9}$

Combine exponents.

Multiply $3$ by $2$ .

$\frac{6 \sqrt[3]{2} \sqrt[3]{3}}{9}$

Combine using the product rule for radicals.

$\frac{6 \sqrt[3]{3 \cdot 2}}{9}$

Multiply $3$ by $2$ .

$\frac{6 \sqrt[3]{6}}{9}$

Step 6

Cancel the common factor of $6$ and $9$ .

Factor $3$ out of $6 \sqrt[3]{6}$ .

$\frac{3 (2 \sqrt[3]{6})}{9}$

Cancel the common factors.

Factor $3$ out of $9$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{2 \sqrt[3]{6}}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{2 \sqrt[3]{6}}{3}$

Decimal Form:

$1.21141372 \dots$

Evaluate cube root of 16/9

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