Problem 26 - Algebra 2

Question

Simplify each radical expression. Use absolute value symbols when needed. $$ \sqrt[3]{27 y^{6}} $$

Step-by-Step Solution

Answer

\(3y^2\)

Step 1: Factor inside the radical

Factor the expression inside the cube root to identify perfect cubes. The number 27 can be expressed as 3 cubed \(3^3\), and \(y^6\) is \(y^2\) cubed because \(y^6 = (y^2)^3\). So the expression inside the radical becomes \(3^3 y^{2\bullet3}\).

Step 2: Apply the cube root

Since the cube root of a product is the product of the cube roots, apply the cube root to each factor: \(\sqrt[3]{3^3}\) and \(\sqrt[3]{y^{2\bullet3}}\). The cube root of any perfect cube is the number that, when cubed, gives the number. Hence, \(\sqrt[3]{3^3} = 3\) and \(\sqrt[3]{y^{2\bullet3}} = y^2\).

Step 3: Write the simplified expression

Combine the results from the previous step to write the simplified expression without a radical: \(3y^2\).

Key Concepts

Cube Root Simplification

Understanding how to simplify cube roots is essential when dealing with radical expressions. A cube root, represented as \(\sqrt[3]{x}\), is a value that, when cubed, gives the original number \(x\). Simplification of cube roots involves factoring the number or expression inside the radical to determine if there are any perfect cubes, which are numbers that can be expressed as the cube of an integer. For example, \(27\) is a perfect cube because it can be written as \(3^3\).

If you encounter an expression under a cube root such as \(\sqrt[3]{27y^6}\), start by factoring the expression to reveal the perfect cubes: \(27\) factors to \(3^3\) and \(y^6\) factors to \(y^2\) cubed. Then apply the cube root to each factor separately: the cube root of \(3^3\) is \(3\), and the cube root of \(y^{2\bullet3}\) is \(y^2\). The final step is to express the result without the radical symbol, yielding a simplified expression. In this case, \(3y^2\).

Factoring Expressions

The process of factoring expressions is breaking down a complicated expression into simpler parts, or factors, that when multiplied together produce the original expression. This is particularly useful when simplifying radical expressions, where identifying and extracting perfect cubes or squares is necessary. For example, factoring the expression \(27y^6\) reveals that \(27\) is a perfect cube, as it is \(3\) cubed, and \(y^6\) can be seen as \(y^2\) cubed.

Effective factoring often relies on recognizing patterns, such as the difference of squares, trinomial patterns, or basic exponential rules. By practicing factoring, you strengthen your ability to manage more complex algebraic expressions and find their simplified forms.

Perfect Cubes

Recognizing perfect cubes is an important skill in simplifying radical expressions. Perfect cubes are numbers that have been cubed from an integer. Some examples of perfect cubes are \(1^3 = 1\), \(2^3 = 8\), \(3^3 = 27\), and \(4^3 = 64\). Knowing these allows for a quick simplification of cube roots.

When you come across a number under a cube root, determine if it is a perfect cube. If it is, simply take the cube root of that number to remove the radical, as with \(\sqrt[3]{27} = 3\). For expressions with variables, remember that a variable to the third power, such as \(y^3\), is also considered a perfect cube. Understanding perfect cubes provides the foundation for working with cube root simplification and other related operations.

Absolute Value

The absolute value of a number is a measure of its distance from zero on the number line, regardless of direction. It's represented by two vertical bars surrounding the number: \(|x|\). When dealing with radicals, especially square roots, absolute value becomes significant because it ensures the non-negative nature of the result. With cube roots, the need for absolute value is less frequent since cubing a negative number results in another negative number.

However, in some cases, particularly when dealing with even exponents under a radical, one might need to use absolute value to address the uncertainty of the sign. For instance, with an expression such as \(\sqrt[3]{y^6}\), where \(y\) could be positive or negative, realizing that \(y^6\) is always positive due to the even exponent is crucial. In this context, incorporating absolute value isn't necessary, but recognizing when it's applicable to an expression affirms your grasp of radicals and their properties.