Simplify ( cube root of 250x^5y^3)/( cube root of 2x^3)

Solution:

Step:1

Merge the cube roots $\sqrt[3]{250x^5{y}^3}$ and $\sqrt[3]{2x^3}$ into one cube root expression: $\sqrt[3]{\frac{250x^5{y}^3}{2x^3}}$.

Step:2

Simplify the fraction $\frac{250x^5{y}^3}{2x^3}$ by removing common factors.

Step:2.1

Extract the factor of $2$ from $250x^5{y}^3$: $\sqrt[3]{\frac{2\cdot 125x^5{y}^3}{2x^3}}$.

Step:2.2

Extract the factor of $2$ from $2x^3$: $\sqrt[3]{\frac{2\cdot 125x^5{y}^3}{2\cdot x^3}}$.

Step:2.3

Eliminate the common factor of $2$: $\sqrt[3]{\frac{\cancel{2}\cdot 125x^5{y}^3}{\cancel{2}{x}^3}}$.

Step:2.4

Restate the simplified expression: $\sqrt[3]{\frac{125x^5{y}^3}{x^3}}$.

Step:3

Remove the common $x$ powers from $x^5$ and $x^3$.

Step:3.1

Factor out $x^3$ from $125x^5{y}^3$: $\sqrt[3]{\frac{x^3\cdot 125x^2{y}^3}{x^3}}$.

Step:3.2

Eliminate the common $x^3$ factors.

Step:3.2.1

Introduce a multiplicative identity of $1$: $\sqrt[3]{\frac{x^3\cdot 125x^2{y}^3}{x^3\cdot 1}}$.

Step:3.2.2

Remove the common $x^3$ factor: $\sqrt[3]{\frac{\cancel{x^3}\cdot 125x^2{y}^3}{\cancel{x^3}\cdot 1}}$.

Step:3.2.3

Rephrase the expression: $\sqrt[3]{\frac{125x^2{y}^3}{1}}$.

Step:3.2.4

Divide $125x^2{y}^3$ by $1$: $\sqrt[3]{125x^2{y}^3}$.

Step:4

Express $125x^2{y}^3$ as $(5y{)}^3{x}^2$.

Step:4.1

Represent $125$ as $5^3$: $\sqrt[3]{5^3{x}^2{y}^3}$.

Step:4.2

Rearrange $x^2$: $\sqrt[3]{5^3{y}^3{x}^2}$.

Step:4.3

Reformulate $5^3{y}^3$ as $(5y{)}^3$: $\sqrt[3]{(5y{)}^3{x}^2}$.

Step:5

Extract terms from the cube root: $5y\sqrt[3]{x^2}$.

Knowledge Notes:

To simplify a radical expression involving cube roots, you can follow these steps:

1. Combine cube roots into a single cube root if you have a fraction under the radical.

2. Factor out common terms in the numerator and denominator.

3. Cancel out common factors.

4. Simplify the expression by reducing powers where possible.

5. If you have a perfect cube under the cube root, you can take it out of the radical.

In this problem, we simplified the cube root of a fraction by canceling out common factors and reducing powers. We also used the property that $\sqrt[3]{a^3}=a$ to simplify the expression further. When dealing with cube roots, it's important to recognize perfect cubes such as $125=5^3$ and to understand the properties of exponents and radicals to simplify expressions correctly.