Answer :
Let's simplify the expression [tex]\(\sqrt[3]{5x} \cdot \sqrt[3]{25x^2}\)[/tex].
To simplify this, we can use the property of exponents which states that the cube root of a product is the product of the cube roots:
[tex]\[
\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b}
\][/tex]
So, let's apply this property:
[tex]\[
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x) \cdot (25x^2)}
\][/tex]
Now, let's calculate the product inside the cube root:
[tex]\[
5x \cdot 25x^2 = 125x^3
\][/tex]
Now we have:
[tex]\[
\sqrt[3]{125x^3}
\][/tex]
The cube root of [tex]\(125x^3\)[/tex] can be found individually for both the number and the variable:
1. The cube root of 125, which is [tex]\(\sqrt[3]{125} = 5\)[/tex] because [tex]\(5^3 = 125\)[/tex].
2. The cube root of [tex]\(x^3\)[/tex], which is [tex]\(\sqrt[3]{x^3} = x\)[/tex].
Therefore, the cube root of the entire expression is:
[tex]\[
\sqrt[3]{125x^3} = 5x
\][/tex]
So, the simplified expression is [tex]\(5x\)[/tex]. The correct answer is [tex]\(5x\)[/tex].
To simplify this, we can use the property of exponents which states that the cube root of a product is the product of the cube roots:
[tex]\[
\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b}
\][/tex]
So, let's apply this property:
[tex]\[
\sqrt[3]{5x} \cdot \sqrt[3]{25x^2} = \sqrt[3]{(5x) \cdot (25x^2)}
\][/tex]
Now, let's calculate the product inside the cube root:
[tex]\[
5x \cdot 25x^2 = 125x^3
\][/tex]
Now we have:
[tex]\[
\sqrt[3]{125x^3}
\][/tex]
The cube root of [tex]\(125x^3\)[/tex] can be found individually for both the number and the variable:
1. The cube root of 125, which is [tex]\(\sqrt[3]{125} = 5\)[/tex] because [tex]\(5^3 = 125\)[/tex].
2. The cube root of [tex]\(x^3\)[/tex], which is [tex]\(\sqrt[3]{x^3} = x\)[/tex].
Therefore, the cube root of the entire expression is:
[tex]\[
\sqrt[3]{125x^3} = 5x
\][/tex]
So, the simplified expression is [tex]\(5x\)[/tex]. The correct answer is [tex]\(5x\)[/tex].
