Simplify the Radical Expression -( cube root of 3)/( cube root of 5)

$- \frac{\sqrt[3]{3}}{\sqrt[3]{5}}$

Step 1

Multiply $\frac{\sqrt[3]{3}}{\sqrt[3]{5}}$ by $\frac{{\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{2}}$ .

$- (\frac{\sqrt[3]{3}}{\sqrt[3]{5}} \cdot \frac{{\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{2}})$

Step 2

Combine and simplify the denominator.

Multiply $\frac{\sqrt[3]{3}}{\sqrt[3]{5}}$ by $\frac{{\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{2}}$ .

$- \frac{\sqrt[3]{3} {\sqrt[3]{5}}^{2}}{\sqrt[3]{5} {\sqrt[3]{5}}^{2}}$

Raise $\sqrt[3]{5}$ to the power of $1$ .

$- \frac{\sqrt[3]{3} {\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{1} {\sqrt[3]{5}}^{2}}$

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

$- \frac{\sqrt[3]{3} {\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{1 + 2}}$

Add $1$ and $2$ .

$- \frac{\sqrt[3]{3} {\sqrt[3]{5}}^{2}}{{\sqrt[3]{5}}^{3}}$

Rewrite ${\sqrt[3]{5}}^{3}$ as $5$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{5}$ as $5^{\frac{1}{3}}$ .

$- \frac{\sqrt[3]{3} {\sqrt[3]{5}}^{2}}{{(5^{\frac{1}{3}})}^{3}}$

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

$- \frac{\sqrt[3]{3} {\sqrt[3]{5}}^{2}}{5^{\frac{1}{3} \cdot 3}}$

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

$- \frac{\sqrt[3]{3} {\sqrt[3]{5}}^{2}}{5^{\frac{3}{3}}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$- \frac{\sqrt[3]{3} {\sqrt[3]{5}}^{2}}{5^{\frac{3}{3}}}$

Rewrite the expression.

$- \frac{\sqrt[3]{3} {\sqrt[3]{5}}^{2}}{5^{1}}$

Evaluate the exponent.

$- \frac{\sqrt[3]{3} {\sqrt[3]{5}}^{2}}{5}$

Step 3

Simplify the numerator.

Rewrite ${\sqrt[3]{5}}^{2}$ as ${(5^{2})}^{\frac{1}{3}}$ .

$- \frac{\sqrt[3]{3} \sqrt[3]{5^{2}}}{5}$

Raise $5$ to the power of $2$ .

$- \frac{\sqrt[3]{3} \sqrt[3]{25}}{5}$

Step 4

Simplify the numerator.

Combine using the product rule for radicals.

$- \frac{\sqrt[3]{3 \cdot 25}}{5}$

Multiply $3$ by $25$ .

$- \frac{\sqrt[3]{75}}{5}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$- \frac{\sqrt[3]{75}}{5}$

Decimal Form:

$- 0.84343266 \dots$

Simplify the Radical Expression -( cube root of 3)/( cube root of 5)

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