Simplify ( cube root of 250x^5y^3)/( cube root of 2x^3)
The problem is asking to perform the operation of division between two cube roots. Specifically, you need to simplify the expression that involves the cube root of the product 250x^5y^3 divided by the cube root of the product 2x^3. This entails simplifying the coefficients (numerical values) and the variables inside the cube roots by applying the laws of exponents and properties of radicals (which in this case are cube roots) to combine and reduce the expression to its simplest form.
$\frac{\sqrt[3]{250 x^{5} y^{3}}}{\sqrt[3]{2 x^{3}}}$
Answer
Solution:
Step:1
Merge the cube roots $\sqrt[3]{250 x^{5} y^{3}}$ and $\sqrt[3]{2 x^{3}}$ into one cube root expression: $\sqrt[3]{\frac{250 x^{5} y^{3}}{2 x^{3}}}$.
Step:2
Simplify the fraction $\frac{250 x^{5} y^{3}}{2 x^{3}}$ by removing common factors.
Step:2.1
Extract the factor of $2$ from $250 x^{5} y^{3}$: $\sqrt[3]{\frac{2 \cdot 125 x^{5} y^{3}}{2 x^{3}}}$.
Step:2.2
Extract the factor of $2$ from $2 x^{3}$: $\sqrt[3]{\frac{2 \cdot 125 x^{5} y^{3}}{2 \cdot x^{3}}}$.
Step:2.3
Eliminate the common factor of $2$: $\sqrt[3]{\frac{\cancel{2} \cdot 125 x^{5} y^{3}}{\cancel{2} x^{3}}}$.
Step:2.4
Restate the simplified expression: $\sqrt[3]{\frac{125 x^{5} y^{3}}{x^{3}}}$.
Step:3
Remove the common $x$ powers from $x^{5}$ and $x^{3}$.
Step:3.1
Factor out $x^{3}$ from $125 x^{5} y^{3}$: $\sqrt[3]{\frac{x^{3} \cdot 125 x^{2} y^{3}}{x^{3}}}$.
Step:3.2
Eliminate the common $x^{3}$ factors.
Step:3.2.1
Introduce a multiplicative identity of $1$: $\sqrt[3]{\frac{x^{3} \cdot 125 x^{2} y^{3}}{x^{3} \cdot 1}}$.
Step:3.2.2
Remove the common $x^{3}$ factor: $\sqrt[3]{\frac{\cancel{x^{3}} \cdot 125 x^{2} y^{3}}{\cancel{x^{3}} \cdot 1}}$.
Step:3.2.3
Rephrase the expression: $\sqrt[3]{\frac{125 x^{2} y^{3}}{1}}$.
Step:3.2.4
Divide $125 x^{2} y^{3}$ by $1$: $\sqrt[3]{125 x^{2} y^{3}}$.
Step:4
Express $125 x^{2} y^{3}$ as $(5 y)^{3} x^{2}$.
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 $(5 y)^{3}$: $\sqrt[3]{(5 y)^{3} x^{2}}$.
Step:5
Extract terms from the cube root: $5 y \sqrt[3]{x^{2}}$.
Knowledge Notes:
To simplify a radical expression involving cube roots, you can follow these steps:
Combine cube roots into a single cube root if you have a fraction under the radical.
Factor out common terms in the numerator and denominator.
Cancel out common factors.
Simplify the expression by reducing powers where possible.
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.
