- The expression contains two like terms, both involving the radical \(\sqrt[5]{x^2 y}\).
- Apply the distributive property: \(4(\sqrt[5]{x^2 y}) + 3(\sqrt[5]{x^2 y}) = (4 + 3)(\sqrt[5]{x^2 y})\).
- Combine the coefficients to get \(7(\sqrt[5]{x^2 y})\).
- The simplified result matches the option \(\boxed{7(\sqrt[5]{x^2 y})}\).
Explanation
- Understanding the Problem
We are given the expression \(4(\sqrt[5]{x^2 y}) + 3(\sqrt[5]{x^2 y})\) and asked to simplify it. Notice that both terms in the sum contain the exact same radical component, \(\sqrt[5]{x^2 y}\). These are known as "like terms" in algebraic expressions. Just as \(4x + 3x = (4+3)x = 7x\), we can use the distributive property of multiplication over addition to combine these terms.
- Applying the Distributive Property
The distributive property states that for any numbers or expressions \(a, b,\) and \(z\), we have \(a \cdot z + b \cdot z = (a + b) \cdot z\). In our case, let \(a = 4\), \(b = 3\), and \(z = \sqrt[5]{x^2 y}\). Substituting these values into the distributive property, we get:
\[4(\sqrt[5]{x^2 y}) + 3(\sqrt[5]{x^2 y}) = (4 + 3)(\sqrt[5]{x^2 y}) = 7(\sqrt[5]{x^2 y})\]
- Identifying the Correct Option
Now that we have simplified the expression to \(7(\sqrt[5]{x^2 y})\), we compare it to the given options:
- \(7(\sqrt[10]{x^2 y})\)
- \(7(\sqrt[10]{x^4 y^2})\)
- \(7(\sqrt[5]{x^2 y})\)
- \(7(\sqrt[5]{x^4 y^2})\)
Our result perfectly matches the third option.
Examples
Understanding how to combine like terms is fundamental in algebra and physics. For example, if you are calculating the total resistance in a circuit where you have multiple components with the same physical properties, such as '4 units of a specific resistance' and '3 units of the same resistance', you can easily determine the total using the same principles used here to combine the radicals. This simplification makes complex equations easier to read, evaluate, and solve in various real-world scientific applications.