Answer :
To simplify the expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex], you need to combine the cube roots by multiplying them under a single cube root:
1. Use the property of radicals: [tex]\(\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b}\)[/tex].
2. Apply this property:
[tex]\[
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x) \cdot (25x^2)}
\][/tex]
3. Multiply the expressions inside the cube root:
[tex]\[
(5x) \cdot (25x^2) = 5 \cdot 25 \cdot x \cdot x^2 = 125x^3
\][/tex]
4. Substitute back into the cube root:
[tex]\[
\sqrt[3]{125x^3}
\][/tex]
5. Simplify the cube root:
[tex]\[
\sqrt[3]{125x^3} = \sqrt[3]{125} \cdot \sqrt[3]{x^3}
\][/tex]
6. Since [tex]\(125 = 5^3\)[/tex], this becomes:
[tex]\[
\sqrt[3]{125} = 5
\][/tex]
7. The cube root of [tex]\(x^3\)[/tex] is simply [tex]\(x\)[/tex]:
[tex]\[
\sqrt[3]{x^3} = x
\][/tex]
8. Combine the results from steps 6 and 7:
[tex]\[
5 \cdot x = 5x
\][/tex]
Therefore, the simplified expression is [tex]\(5x\)[/tex].
1. Use the property of radicals: [tex]\(\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b}\)[/tex].
2. Apply this property:
[tex]\[
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x) \cdot (25x^2)}
\][/tex]
3. Multiply the expressions inside the cube root:
[tex]\[
(5x) \cdot (25x^2) = 5 \cdot 25 \cdot x \cdot x^2 = 125x^3
\][/tex]
4. Substitute back into the cube root:
[tex]\[
\sqrt[3]{125x^3}
\][/tex]
5. Simplify the cube root:
[tex]\[
\sqrt[3]{125x^3} = \sqrt[3]{125} \cdot \sqrt[3]{x^3}
\][/tex]
6. Since [tex]\(125 = 5^3\)[/tex], this becomes:
[tex]\[
\sqrt[3]{125} = 5
\][/tex]
7. The cube root of [tex]\(x^3\)[/tex] is simply [tex]\(x\)[/tex]:
[tex]\[
\sqrt[3]{x^3} = x
\][/tex]
8. Combine the results from steps 6 and 7:
[tex]\[
5 \cdot x = 5x
\][/tex]
Therefore, the simplified expression is [tex]\(5x\)[/tex].
