To simplify the given expression [tex]\(\sqrt[3]{27 x y^2} + \sqrt[3]{8 x^4 y^5}\)[/tex], let's break it down in steps and simplify each term.
### Step 1: Simplify [tex]\(\sqrt[3]{27 x y^2}\)[/tex]
First, we notice that [tex]\(27\)[/tex] is a perfect cube:
[tex]\[27 = 3^3\][/tex]
So, the cube root of [tex]\(27\)[/tex] is:
[tex]\[\sqrt[3]{27} = 3\][/tex]
Next, let’s separate the variables under the cube root:
[tex]\[\sqrt[3]{27 x y^2} = \sqrt[3]{27} \cdot \sqrt[3]{x} \cdot \sqrt[3]{y^2}\][/tex]
Combine what we know:
[tex]\[\sqrt[3]{27 x y^2} = 3 \cdot \sqrt[3]{x} \cdot \sqrt[3]{y^2}\][/tex]
Simplify the cube roots:
[tex]\[\sqrt[3]{y^2} = y^{2/3}\][/tex]
[tex]\[\sqrt[3]{x} = x^{1/3}\][/tex]
Putting it all together:
[tex]\[\sqrt[3]{27 x y^2} = 3 \cdot x^{1/3} \cdot y^{2/3}\][/tex]
### Step 2: Simplify [tex]\(\sqrt[3]{8 x^4 y^5}\)[/tex]
Next, we notice that [tex]\(8\)[/tex] is also a perfect cube:
[tex]\[8 = 2^3\][/tex]
So, the cube root of [tex]\(8\)[/tex] is:
[tex]\[\sqrt[3]{8} = 2\][/tex]
Now, let’s separate the variables under the cube root:
[tex]\[\sqrt[3]{8 x^4 y^5} = \sqrt[3]{8} \cdot \sqrt[3]{x^4} \cdot \sqrt[3]{y^5}\][/tex]
Combine what we know:
[tex]\[\sqrt[3]{8 x^4 y^5} = 2 \cdot \sqrt[3]{x^4} \cdot \sqrt[3]{y^5}\][/tex]
Simplify the cube roots:
[tex]\[\sqrt[3]{x^4} = x^{4/3}\][/tex]
[tex]\[\sqrt[3]{y^5} = y^{5/3}\][/tex]
Putting it all together:
[tex]\[\sqrt[3]{8 x^4 y^5} = 2 \cdot x^{4/3} \cdot y^{5/3}\][/tex]
### Step 3: Combine the simplified terms
Now combine both results:
[tex]\[\sqrt[3]{27 x y^2} + \sqrt[3]{8 x^4 y^5} = 3 x^{1/3} y^{2/3} + 2 x^{4/3} y^{5/3}\][/tex]
Thus, the simplified form of the given expression is:
[tex]\[3 x^{1/3} y^{2/3} + 2 x^{4/3} y^{5/3}\][/tex]
This is the step-by-step solution to simplifying the given expression.
