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Answer :
- Combine the cube roots using the property $\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab}$.
- Simplify the expression inside the cube root: $(5x)(25x^2) = 125x^3$.
- Rewrite the expression as $\sqrt[3]{125x^3}$.
- Simplify the cube root: $\sqrt[3]{125x^3} = 5x$. 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 can 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
Now, we simplify the expression inside the cube root:
$$(5x)(25x^2) = 5 \cdot 25 \cdot x \cdot x^2 = 125x^3$$
4. Rewriting the Expression
So, we have $\sqrt[3]{125x^3}$. Since $125 = 5^3$, we can rewrite the expression as:
$$\sqrt[3]{125x^3} = \sqrt[3]{5^3x^3}$$
5. Simplifying the Cube Root
Now, we take the cube root of $5^3x^3$:
$$\sqrt[3]{5^3x^3} = 5x$$
6. Final Answer
Therefore, the simplified expression is $5x$.
### Examples
Imagine you are calculating the volume of a storage container. If the dimensions involve cube roots, simplifying expressions like this helps in determining the overall volume more easily. Understanding how to simplify radical expressions is useful in various fields, including engineering and physics, where complex calculations are common.
- Simplify the expression inside the cube root: $(5x)(25x^2) = 125x^3$.
- Rewrite the expression as $\sqrt[3]{125x^3}$.
- Simplify the cube root: $\sqrt[3]{125x^3} = 5x$. 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 can 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
Now, we simplify the expression inside the cube root:
$$(5x)(25x^2) = 5 \cdot 25 \cdot x \cdot x^2 = 125x^3$$
4. Rewriting the Expression
So, we have $\sqrt[3]{125x^3}$. Since $125 = 5^3$, we can rewrite the expression as:
$$\sqrt[3]{125x^3} = \sqrt[3]{5^3x^3}$$
5. Simplifying the Cube Root
Now, we take the cube root of $5^3x^3$:
$$\sqrt[3]{5^3x^3} = 5x$$
6. Final Answer
Therefore, the simplified expression is $5x$.
### Examples
Imagine you are calculating the volume of a storage container. If the dimensions involve cube roots, simplifying expressions like this helps in determining the overall volume more easily. Understanding how to simplify radical expressions is useful in various fields, including engineering and physics, where complex calculations are common.
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