Problem 46 - Algebra 2

Question

Simplify each radical expression. Use absolute value symbols when needed. $$ \sqrt[5]{y^{20}} $$

Step-by-Step Solution

Answer

The simplified form of the expression \(\sqrt[5]{y^{20}}\) is \(y^4\).

Step 1: Understand the expression

The expression \(\sqrt[5]{y^{20}}\) represents the fifth root of y to the power of 20. This basically asks the question: 'What number, when multiplied by itself five times, gives y to the power of 20?' We need to find that number.

Step 2: Simplify the expression

Since y to the power of 20 is the same as \((y^4)^5\), the fifth root of y to the power of 20 is simply y to the power of 4, or \(y^4\). This is because the base (y) is the same and the powers (4 and 5) multiply to give 20.

Step 3: Consider the need for absolute value symbols

Since there is no square root, cube root, or any other even root, there will not be a scenario where the result could be negative. Therefore, there is no need for absolute value symbols in this case.

Key Concepts

Fifth Roots

A fifth root is all about finding the number that, when multiplied by itself five times, equals the given value under the radical. To find the fifth root of a number, you are essentially asking, ‘What number raised to the power of 5 will give me this value?’
Fifth roots can often seem tricky, but they follow the same basic principles of finding say, a square or cube root.

• When dealing with expressions like \(\sqrt[5]{y^{20}}\), it means you want to determine a number which, when used five times in a multiplication, yields \(y^{20}\).
• By recognizing \(y^{20}\) as \((y^4)^5\), you simplify the fifth root to \(y^4\).
• This simplification is a common technique when tackling roots of powers, where you express the power as a multiple of the root's degree.

Learning to simplify such expressions helps in dealing with more complex equations and expressions effectively.

Properties of Exponents

Understanding the properties of exponents is crucial for simplifying expressions that contain powers and roots. When you see something like \(y^{20}\), knowing how to manipulate this expression using exponent rules is key. Here are a few fundamental properties:

• Power of a Power Rule: This rule states that \((a^m)^n = a^{m \cdot n}\). You use this property to recognize \(y^{20}\) as \((y^4)^5\).
• Product of Powers Rule: This states that \(a^m \cdot a^n = a^{m+n}\). When simplifying expressions, combining powers of similar bases can be a handy tool.
• Root and Exponent Relationship: When you apply a root to an exponent, such as a fifth root to \(y^{20}\), you can express it as \(y^{20/5}\). This is exactly what happens when we express it as \(y^4\).

Mastering these properties means you'll be ready to tackle any exponent-related challenge with confidence.

Absolute Value Symbols

Absolute value symbols often appear in radical expressions where an even root might result in both positive and negative values. However, in our problem, we are dealing with a fifth root, which is an odd root.
When working with odd roots, there is no need for absolute value symbols:

• Odd roots, such as the fifth root in this expression, will have only one principal root, which can be positive or negative naturally.
• Since there's no scenario where the outcome of a fifth root requires a correction for positivity, you don't need absolute value signs to simplify the result.
• In this case, simplifying \(\sqrt[5]{y^{20}}\) results in \(y^4\) without any need for these symbols.

Absolute value symbols are more relevant when dealing with even roots, as they require ensuring that the outcome reflects a principal non-negative value.