Problem 75 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 57-80: Simplify the expression, and rationalize the denominator when appropriate. $$ \sqrt[4]{\left(3 x^{5} y^{-2}\right)^{4}} $$

Step-by-Step Solution

Answer

The simplified expression is \(3x^5y^{-2}\).

Step 1: Apply the Power Rule

The expression inside the fourth root is \((3x^5y^{-2})^4\). By the power rule \((a^m)^n = a^{m \times n}\), simplify the expression to \(3^4x^{5 \cdot 4}y^{-2 \cdot 4}\).

Step 2: Calculate the Powers

Now calculate each power: - \(3^4 = 81\)- \(x^{20} = x^{5 \cdot 4}\)- \(y^{-8} = y^{-2 \cdot 4}\)The simplified expression becomes \(81x^{20}y^{-8}\).

Step 3: Apply the Fourth Root

We need to take the fourth root of the expression \(\sqrt[4]{81x^{20}y^{-8}}\). This means applying the fourth root separately to each factor: \(\sqrt[4]{81}\times\sqrt[4]{x^{20}}\times\sqrt[4]{y^{-8}}\).

Step 4: Simplify Each Factor's Root

Simplify each term:- \(\sqrt[4]{81} = 3\) because \(3^4 = 81\).- \(\sqrt[4]{x^{20}} = x^{5}\) since \(x^{20 \div 4} = x^5\).- \(\sqrt[4]{y^{-8}} = y^{-2}\) since \(y^{-8 \div 4} = y^{-2}\).

Step 5: Write the Final Expression

Putting it all together, the simplified expression is \(3x^5y^{-2}\). We don't need to rationalize the denominator as there is no denominator involved.

Key Concepts

Power Rule

The power rule is a fundamental concept in algebra that helps simplify expressions with exponents by considering the relationship between exponents. When you have a term raised to a power, and that entire term is once again raised to another power, the power rule provides a shortcut for simplification. The basic formula is \((a^m)^n = a^{m \times n}\).
This means you multiply the exponents together. This rule simplifies the process by reducing the number of steps needed to work with complex exponent expressions.
For example, with \((3x^5y^{-2})^4\), you can quickly apply the power rule to get each term to its proper power:

• The constant \(3\) becomes \(3^4\), resulting in \(81\).
• The variable \(x^5\) becomes \(x^{20}\) since \(5 \times 4 = 20\).
• The variable \(y^{-2}\) turns into \(y^{-8}\) as \(-2 \times 4 = -8\).

This simplification is powerful because the calculation becomes straightforward once you multiply the exponents.

Fourth Root

Understanding the concept of the fourth root can make simplifying expressions much easier. The fourth root of a number is a value that, when multiplied by itself four times, returns the original number.
It's denoted as \(\sqrt[4]{a}\). In other words, to find the fourth root of a number, you are looking for a number that raises to the fourth power to equal your original number.
In the given exercise, you need to find the fourth root of the simplified expression \(81x^{20}y^{-8}\). You do this by addressing each component separately:

• The fourth root of \(81\) is \(3\), because \(3^4 = 81\).
• The fourth root of \(x^{20}\) can be calculated by dividing the exponent by \(4\), resulting in \(x^{5}\).
• The fourth root of \(y^{-8}\) is \(y^{-2}\), derived from \(-8/4 = -2\).

By treating each term individually, you can handle even complex expressions systematically. This makes the fourth root concept a critical tool for tackling problems like this one.

Exponent Rules

Exponent rules are foundational in simplifying expressions involving powers. They provide a set of guidelines that dictate how to handle numbers raised to powers.
The key exponent rules involved in this exercise include:

• Product Rule: \(a^m \times a^n = a^{m+n}\)
• Quotient Rule: \(\frac{a^m}{a^n} = a^{m-n}\)
• Power Rule: \((a^m)^n = a^{m\times n}\)
• Negative Exponent Rule: \(a^{-n} = \frac{1}{a^n}\)

These rules allow for quick manipulation and simplification of expressions.
In our exercise, the power rule and understanding of negative exponents are prominently used.\(x^{5 \times 4} = x^{20}\) by multiplying the exponents, and \(y^{-2 \times 4} = y^{-8}\), showcasing the application of these rules effectively to simplify the given expression.
By remembering these basic rules, solving problems involving exponents becomes more straightforward and manageable.

Simplification Steps

Simplifying mathematical expressions can seem daunting. However, with a systematic approach using known rules and steps, it becomes manageable. The original expression \(\sqrt[4]{(3x^5y^{-2})^4}\) appears complex, but breaking it into steps makes it clearer.
Here's how you approach it:

• First, use the Power Rule to simplify the expression inside the root: move \((3x^5y^{-2})^4\) to \(81x^{20}y^{-8}\).
• Next, focus on the Fourth Root: apply it separately to each term.
• Utilize your understanding of exponent rules at each step to see: \(\sqrt[4]{81} = 3\), \(\sqrt[4]{x^{20}} = x^5\), and \(\sqrt[4]{y^{-8}} = y^{-2}\).
• Finally, combine these simplified components to present the simplified expression: \(3x^5y^{-2}\). No need for rationalization, as there is no denominator present.

Each of these steps builds on the concepts we've discussed (power rule, fourth root, exponent rules), and demonstrates their practical use in simplifying expressions.