Answer :
To solve the expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex], we first need to simplify each cube root and then multiply them together.
1. Simplify [tex]\(\sqrt[3]{5x}\)[/tex]:
- This term is already in its simplest form.
2. Simplify [tex]\(\sqrt[3]{25x^2}\)[/tex]:
- Let's break this down further. The expression inside the cube root is [tex]\(25x^2\)[/tex], which can be factored as [tex]\((5^2) \cdot x^2\)[/tex].
- When taking the cube root of a product, you can apply the cube root to each factor separately: [tex]\(\sqrt[3]{(5^2) \cdot x^2}\)[/tex].
- This becomes [tex]\(\sqrt[3]{5^2} \cdot \sqrt[3]{x^2}\)[/tex].
3. Calculate individual cube roots:
- [tex]\(\sqrt[3]{5^2} = \sqrt[3]{25}\)[/tex]. Since 25 doesn’t have a cube root that simplifies nicely, we leave it as [tex]\(\sqrt[3]{25}\)[/tex].
- [tex]\(\sqrt[3]{x^2}\)[/tex] is already simplified.
4. Combine the simplified terms:
- Now, multiply [tex]\(\sqrt[3]{5x}\)[/tex] and [tex]\(\sqrt[3]{25x^2}\)[/tex]: [tex]\(\sqrt[3]{5x} \cdot \left(\sqrt[3]{25} \cdot \sqrt[3]{x^2}\right)\)[/tex].
- By properties of cube roots, this becomes [tex]\(\sqrt[3]{5x \cdot 25x^2} = \sqrt[3]{125x^3}\)[/tex].
5. Simplify [tex]\(\sqrt[3]{125x^3}\)[/tex]:
- Notice that [tex]\(125 = 5^3\)[/tex] and [tex]\(x^3\)[/tex] is a perfect cube.
- So, [tex]\(\sqrt[3]{125x^3} = \sqrt[3]{(5^3) \cdot x^3} = 5x\)[/tex].
Thus, the simplified form of [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex] is [tex]\(5x\)[/tex]. The answer is:
[tex]\[ 5x \][/tex]
1. Simplify [tex]\(\sqrt[3]{5x}\)[/tex]:
- This term is already in its simplest form.
2. Simplify [tex]\(\sqrt[3]{25x^2}\)[/tex]:
- Let's break this down further. The expression inside the cube root is [tex]\(25x^2\)[/tex], which can be factored as [tex]\((5^2) \cdot x^2\)[/tex].
- When taking the cube root of a product, you can apply the cube root to each factor separately: [tex]\(\sqrt[3]{(5^2) \cdot x^2}\)[/tex].
- This becomes [tex]\(\sqrt[3]{5^2} \cdot \sqrt[3]{x^2}\)[/tex].
3. Calculate individual cube roots:
- [tex]\(\sqrt[3]{5^2} = \sqrt[3]{25}\)[/tex]. Since 25 doesn’t have a cube root that simplifies nicely, we leave it as [tex]\(\sqrt[3]{25}\)[/tex].
- [tex]\(\sqrt[3]{x^2}\)[/tex] is already simplified.
4. Combine the simplified terms:
- Now, multiply [tex]\(\sqrt[3]{5x}\)[/tex] and [tex]\(\sqrt[3]{25x^2}\)[/tex]: [tex]\(\sqrt[3]{5x} \cdot \left(\sqrt[3]{25} \cdot \sqrt[3]{x^2}\right)\)[/tex].
- By properties of cube roots, this becomes [tex]\(\sqrt[3]{5x \cdot 25x^2} = \sqrt[3]{125x^3}\)[/tex].
5. Simplify [tex]\(\sqrt[3]{125x^3}\)[/tex]:
- Notice that [tex]\(125 = 5^3\)[/tex] and [tex]\(x^3\)[/tex] is a perfect cube.
- So, [tex]\(\sqrt[3]{125x^3} = \sqrt[3]{(5^3) \cdot x^3} = 5x\)[/tex].
Thus, the simplified form of [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex] is [tex]\(5x\)[/tex]. The answer is:
[tex]\[ 5x \][/tex]
