Answer :
- Combine the cube roots using the property $\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab}$: $\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x)(25x^2)}$.
- Simplify the expression inside the cube root: $(5x)(25x^2) = 125x^3$.
- Take the cube root: $\sqrt[3]{125x^3} = \sqrt[3]{125} \cdot \sqrt[3]{x^3}$.
- Evaluate the cube roots to get the final simplified expression: $5x$. The final answer is $\boxed{5x}$.
### Explanation
1. Understanding the Problem
We are given the expression $\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}$ and asked to simplify it completely. This involves combining the cube roots and simplifying the resulting expression.
2. Combining Cube Roots
To simplify the expression, we use the property that $\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab}$. Applying this property, we have:
$$\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x)(25x^2)}$$
3. Simplifying the Expression Inside the Cube Root
Now, we simplify the expression inside the cube root:
$$(5x)(25x^2) = 5 \cdot 25 \cdot x \cdot x^2 = 125x^3$$
4. Taking the Cube Root
So, we have:
$$\sqrt[3]{125x^3}$$
Now, we take the cube root of $125x^3$. We know that $\sqrt[3]{125} = 5$ and $\sqrt[3]{x^3} = x$. Therefore,
$$\sqrt[3]{125x^3} = \sqrt[3]{125} \cdot \sqrt[3]{x^3} = 5x$$
5. Final Answer
Thus, the simplified expression is $5x$.
### Examples
Imagine you are calculating the volume of two cubes with sides expressed as cube roots. Simplifying expressions like this helps in determining the combined volume or comparing the sizes of the cubes more easily. This type of simplification is useful in various engineering and physics applications where expressions involving radicals and variables need to be handled efficiently. Understanding how to manipulate these expressions allows for easier calculations and better insights into the relationships between different quantities.
- Simplify the expression inside the cube root: $(5x)(25x^2) = 125x^3$.
- Take the cube root: $\sqrt[3]{125x^3} = \sqrt[3]{125} \cdot \sqrt[3]{x^3}$.
- Evaluate the cube roots to get the final simplified expression: $5x$. The final answer is $\boxed{5x}$.
### Explanation
1. Understanding the Problem
We are given the expression $\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}$ and asked to simplify it completely. This involves combining the cube roots and simplifying the resulting expression.
2. Combining Cube Roots
To simplify the expression, we use the property that $\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab}$. Applying this property, we have:
$$\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x)(25x^2)}$$
3. Simplifying the Expression Inside the Cube Root
Now, we simplify the expression inside the cube root:
$$(5x)(25x^2) = 5 \cdot 25 \cdot x \cdot x^2 = 125x^3$$
4. Taking the Cube Root
So, we have:
$$\sqrt[3]{125x^3}$$
Now, we take the cube root of $125x^3$. We know that $\sqrt[3]{125} = 5$ and $\sqrt[3]{x^3} = x$. Therefore,
$$\sqrt[3]{125x^3} = \sqrt[3]{125} \cdot \sqrt[3]{x^3} = 5x$$
5. Final Answer
Thus, the simplified expression is $5x$.
### Examples
Imagine you are calculating the volume of two cubes with sides expressed as cube roots. Simplifying expressions like this helps in determining the combined volume or comparing the sizes of the cubes more easily. This type of simplification is useful in various engineering and physics applications where expressions involving radicals and variables need to be handled efficiently. Understanding how to manipulate these expressions allows for easier calculations and better insights into the relationships between different quantities.
