Certainly! Let's solve the problem step-by-step.
We are given the expression:
[tex]\[ \sqrt[3]{x^2 y^4} \cdot \sqrt[3]{x^2 y} \][/tex]
Our goal is to simplify this product of cube roots.
### Step 1: Rewrite the Cube Roots
First, we rewrite both terms under a common cube root:
[tex]\[ \sqrt[3]{x^2 y^4} \cdot \sqrt[3]{x^2 y} = \sqrt[3]{(x^2 y^4) \cdot (x^2 y)} \][/tex]
### Step 2: Combine the Terms Under the Cube Root
Next, we multiply the expressions inside the cube root:
[tex]\[ (x^2 y^4) \cdot (x^2 y) = x^2 \cdot x^2 \cdot y^4 \cdot y = x^{2+2} y^{4+1} = x^4 y^5 \][/tex]
So the expression becomes:
[tex]\[ \sqrt[3]{x^4 y^5} \][/tex]
### Step 3: Simplify the Cube Root
Now, let's look at the simplified expression under the cube root. Separating the terms gives us:
[tex]\[ \sqrt[3]{x^4 y^5} = (\sqrt[3]{x^4}) \cdot (\sqrt[3]{y^5}) \][/tex]
We can further simplify each term individually:
[tex]\[ \sqrt[3]{x^4} = x^{4/3} \][/tex]
[tex]\[ \sqrt[3]{y^5} = y^{5/3} \][/tex]
Thus, combining these results, we get:
[tex]\[ x^{4/3} \cdot y^{5/3} \][/tex]
### Step 4: Restate with Decimal Exponents
To write the exponents in decimal form:
[tex]\[ x^{4/3} = x^{1.333333333333333} \][/tex]
[tex]\[ y^{5/3} = y^{1.666666666666667} \][/tex]
Hence, the final expression is:
[tex]\[ (x^{1.333333333333333}) \cdot (y^{1.666666666666667}) \][/tex]
Therefore, the simplified form of the given expression is:
[tex]\[ \sqrt[3]{x^2 y^4} \cdot \sqrt[3]{x^2 y} = (x^{1.333333333333333}) \cdot (y^{1.666666666666667}) \][/tex]
Or alternatively, you could express it as:
[tex]\[ (x^2 y)^{1/3} \cdot (x^2 y^4)^{1/3} \][/tex]
Which matches our original result.
