Problem 23 - Algebra 2

Question

Simplify. \(\sqrt[3]{16 y^{3}}\)

Step-by-Step Solution

Answer

The simplified form is \(2y\sqrt[3]{2}\).

Step 1: Factor the Radicand

First, break down the number and variable inside the cube root. The expression inside is \(16y^3\). Notice that \(16\) can be factored as \(2^4\), and \(y^3\) is already a cube. Thus, we can rewrite the radicand as \(2^4y^3\).

Step 2: Separate into Separate Radicals

Apply the property of radicals that allows us to separate the terms inside the cube root: \( \sqrt[3]{2^4y^3} = \sqrt[3]{2^4} \cdot \sqrt[3]{y^3} \). This helps in simplifying each part separately.

Step 3: Simplify the Variable Part

Focus on \(\sqrt[3]{y^3}\). Since the cube root and the exponent are the same, they cancel out. Therefore, \(\sqrt[3]{y^3} = y\).

Step 4: Simplify the Numerical Part

Now, simplify \(\sqrt[3]{2^4}\). Write \(2^4\) as \(2^3 \cdot 2\). Apply the property \(\sqrt[3]{a^3} = a\), thus \(\sqrt[3]{2^3} = 2\). What remains inside the cube root is \(\sqrt[3]{2}\).

Step 5: Combine the Simplified Parts

Combine the simplified results of the numerical and variable parts: \(2y\sqrt[3]{2}\). There is no further simplification to be done. This is the final simplified form of the expression.

Key Concepts

Cube Root

The cube root is a special type of root in mathematics. It helps us determine what number, when multiplied by itself three times, will result in the given number. If we are finding the cube root of a given expression like \( \sqrt[3]{16y^3} \), our goal is to split it in a way to identify any perfect cubes. This is similar to finding square roots but instead, it involves the 'cube' rather than the 'square'.

For instance, the cube root of \( 8 \) is \( 2 \), because \( 2 \times 2 \times 2 = 8 \). Similarly, for the variable \( y^3 \), the cube root takes it back to \( y \), since \( y \times y \times y = y^3 \). Understanding how to identify and simplify cube roots is crucial when working with radicals.

Properties of Radicals

Radicals, like roots, follow certain properties which help in their simplification. One of the main properties is that a radical can be distributed over multiplication. This means that \( \sqrt[n]{a \cdot b} = \sqrt[n]{a} \cdot \sqrt[n]{b} \).

For cube roots, we apply the property as used in this problem: \( \sqrt[3]{2^4y^3} = \sqrt[3]{2^4} \cdot \sqrt[3]{y^3} \). By splitting the radical, we can easily simplify each part of the expression separately, which makes simplifying complex radicals much easier. These principles are invaluable in algebra when dealing with larger expressions that need simplification.

Factoring Radicands

Factoring the radicand is the first vital step in simplifying radicals. A radicand is the expression inside a root. For example, in \( \sqrt[3]{16y^3} \), the radicand is \( 16y^3 \). The process requires identifying any perfect cube components:

• Break down numbers into their prime factors; for 16, this is \( 2^4 \).
• Recognize the cubed components in variables; \( y^3 \) is already a perfect cube.

This enables us to restructure the radicand and facilitate its separation into simpler cubes that are more manageable to handle together with their associated cube roots.

Simplifying Expressions

Simplifying expressions involving radicals, especially cube roots, combines multiple steps to achieve the simplest form. Once we've broken down the radicand, and applied properties of radicals, we proceed to simplify each radical individually. For this example, follow these steps:

• Simplify the cube root of the variable portion by cancelling the power with the root: \( \sqrt[3]{y^3} = y \).
• For the numerical part, \( 2^4 \) is broken into \( 2^3 \cdot 2 \), simplifying \( \sqrt[3]{2^3} = 2 \).

Finally, combine the simplified components into the final expression. The result here is \( 2y\sqrt[3]{2} \). Proper factoring and splitting make this process a lot more straightforward, enabling even complex expressions to be untangled into simpler, more usable forms.