Question

Simplify the expression.\(\sqrt[3]{5} \cdot \sqrt[3]{5^{2}}\)

Short answer

The simplified expression is 5.

Step-by-step solution

Identify the radicands and the root

In this exercise, the radicands are 5 and \(5^{2}\), and the root is 3. The expression can be written as \(\sqrt[3]{5} \cdot \sqrt[3]{5^{2}}\).

Apply the multiplication property of radicals

Multiplication property of radicals states that two radicals with the same root can be simplified by multiplying the radicands and taking the root of the product. Applying this rule gives us: \(\sqrt[3]{5 \cdot 5^{2}}\)

Simplify the multiplication inside the radical

Perform the multiplication inside the root to obtain \(\sqrt[3]{5^{3}}\)

Simplify the Cube Root

The cube root of \(5^{3}\) is simply 5. Thus the simplified expression is 5.

Key concepts

Understanding Radicands

The term radicand refers to the number or expression inside a radical sign. In the context of cube roots, the radicand is the value that we are taking the third root of. For instance, in the expression \(\sqrt[3]{5}\), 5 is the radicand. It is essential to recognize the radicand because it is the number which we are essentially breaking down to its root.

In the given exercise, we have two radicands, 5 and \(5^{2}\), both under cube root signs. Dealing with radicands correctly is important as it allows us to harness the properties of radicals—such as multiplication or simplification—to simplify expressions. Making a mistake in identifying the radicand can lead to incorrect simplification and thus, wrong answers.

Applying Root Operations

Root operations involve finding the root of a number, which in our case is a cube root, indicating the inverse operation of raising a number to the power of 3. The cube root of a number 'x' is a number 'y' such that \(y^3 = x\). Understanding how to manage these operations is crucial when simplifying radical expressions.

For the exercise \(\sqrt[3]{5} \cdot \sqrt[3]{5^{2}}\), recognizing that both terms under the radical sign are cube roots (\(\sqrt[3]{ }\)) means you can use specific rules for simplifying. It's important to become comfortable with these operations to successfully manipulate and simplify radical expressions effectively.

Multiplication Property of Radicals

The multiplication property of radicals is a key tool when simplifying expressions involving roots. It states that if you have two radicals with the same type of root, you can multiply the radicands together under a single radical sign. In mathematical terms, \(\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \times b}\). This property works with any index—not just square roots but cube roots and others as well.

Applying this property to our original problem, \(\sqrt[3]{5} \cdot \sqrt[3]{5^{2}}\) allows us to combine the two cube roots into one: \(\sqrt[3]{5 \cdot 5^{2}}\). Understanding this property is crucial for simplification because it reduces the number of radical expressions and makes the problem much more manageable. Additionally, it paves the way to further simplification by setting up the expression for the potential extraction of perfect roots.