Problem 71 - Algebra 2

Question

Simplify each radical expression. Use absolute value bars where they are needed. $$ \sqrt{x^{10} y^{100}} $$

Step-by-Step Solution

Answer

The simplified form of the radical expression \(\sqrt{x^{10} y^{100}}\) is \(|x^{5}| y^{50}\).

Step 1: Calculate the square root of each variable independently

The variables \(x\) and \(y\) do not multiply or divide by one another, thus it is feasible to square root each term independently. Therefore, the square root of \(x^{10}\) is calculated first, followed by the square root of \(y^{100}\).

Step 2: Apply the rule \(\sqrt{x^n} = x^{n/2}\)

With the help of the rule \(\sqrt{x^n} = x^{n/2}\), the square root of \(x^{10}\) and \(y^{100}\) can be computed as follows: \(\sqrt{x^{10}} = x^{10/2} = x^{5}, \sqrt{y^{100}} = y^{100/2}=y^{50}\)

Step 3: Take into account the absolute value bars

Since the square root of a number squared is the absolute value of that number, the square root of \(x^{10}\) and \(y^{100}\) will yield \(|x^{5}|\) and \(y^{50}\), respectively. However, as \(y^{50}\) (50 is an even number) is always positive, it does not need absolute value bars.

Step 4: Combine the results

The radical expression simplifies to \(|x^{5}| y^{50}\)

Key Concepts

Simplifying Radicals

The beauty of simplifying radicals lies in turning complex expressions into simpler forms. Let's break down what it means to simplify a radical expression. Radicals often involve expressions under a square root or other roots. When simplifying, you aim to remove the radical when possible, making the expression cleaner and easier to work with.

To simplify a radical:

• Look for perfect squares, cubes, etc., that can be extracted from the radical.
• Apply mathematical rules that allow you to operate directly on the numbers or variables inside the radical.

This can dramatically reduce the complexity of an expression. The goal is to present the radical in its simplest form while still being mathematically equivalent.

Absolute Value

Absolute value is all about measuring the distance from zero on a number line, which means it is always non-negative. In mathematical terms, the absolute value of a number is represented by vertical bars like this: \(|x|\).

When simplifying radicals, absolute values become crucial because:

• They ensure expressions involving even roots remain non-negative.
• If a variable raised to an odd power comes from an even power root, you might need absolute value bars to reflect that it can be either negative or positive.

In the example, \[\sqrt{x^{10}} = x^{5}\] requires absolute value bars to ensure it stays non-negative, thus \(|x^5|\). This reinforces that under square roots, what you end up with must be either zero or positive.

Square Roots

Square roots are used to determine what number, when multiplied by itself, will give you the original number. For instance, the square root of 9 is 3 because \(3 \times 3 = 9\).

For variables, when you take the square root of something like \(x^{10}\), you're seeking what power of \(x\) gives value when squared:

• Apply the rule \(\sqrt{x^n} = x^{n/2}\).
• This means \(\sqrt{x^{10}} = x^{10/2} = x^5\).

This method helps in breaking down expressions involving variables and exponents under a square root into simpler forms, making them easier to manage and understand.

Exponent Rules

Exponents are shorthand for repeated multiplication. They play an essential part in simplifying radical expressions. Understanding the exponent rules helps streamline calculations and find equivalent expressions.

Some primary exponent rules include:

• \(a^m \times a^n = a^{m+n}\)
• \((a^m)^n = a^{m \times n}\)
• \(a^{-n} = \frac{1}{a^n}\)

In our exercise, using the rule \(\sqrt{x^n} = x^{n/2}\), we efficiently simplify \(x^{10}\) and \(y^{100}\) by halving their exponents. Recognizing and applying these rules make the process of handling and reducing exponents in radical expressions straightforward and manageable.