Problem 79 - College Algebra with Modeling and Visualization

Question

Simplify the expression and write it with rational exponents. Assume that all variables are positive. $$ \left(x^{2} y^{8}\right)^{1 / 2} $$

Step-by-Step Solution

Answer

The simplified expression is \( x y^4 \).

Step 1: Apply Exponent Rule

When you have a power raised to another power, the exponents multiply. Here, the expression \((x^2 y^8)^{1/2}\) follows this rule. Therefore, apply the exponent rule: \( (a^m)^n = a^{m imes n} \).

Step 2: Distribute the Exponent

Apply the \( \frac{1}{2} \) exponent to each factor inside the parentheses: \( (x^2)^{1/2} \) and \( (y^8)^{1/2} \). This separates the expression into two parts that can be individually simplified.

Step 3: Simplify Each Element

Simplify \( (x^2)^{1/2} \): Using the exponent rule, \((x^2)^{1/2} = x^{2 imes 1/2} = x^1 = x\).

Step 4: Simplify Second Element

Simplify \((y^8)^{1/2}\): Use the exponent rule, \((y^8)^{1/2} = y^{8 imes 1/2} = y^4\).

Step 5: Combine Simplified Elements

Combine the results from the previous steps: \( x imes y^4 \). This is the simplified expression with rational exponents.

Key Concepts

Exponent Rules

Understanding exponent rules is crucial when working with expressions that include powers. These rules simplify complex expressions by breaking down how we deal with exponents. When we have an expression like \[ (x^2 y^8)^{1/2} \], it involves a key exponent rule: when you have a power raised to another power, you multiply the exponents. This can be expressed as:

• \[ (a^m)^n = a^{m \times n} \]

This rule helps us compress multiple layers of exponents into single terms that are much easier to manage.
By applying this rule to \[ (x^2 y^8)^{1/2} \], we first distribute the \( \frac{1}{2} \) exponent to both \( x^2 \) and \( y^8 \). This means each factor inside the parentheses needs simplification on its own, and then the results can be combined for the final simplified expression.

Simplifying Expressions

Simplifying expressions is a method of making mathematical expressions less complex and easier to work with. For the expression \[ (x^2 y^8)^{1/2} \], the goal is to rewrite it in a more straightforward form using rational exponents. Here are the steps needed to simplify the expression:

• Distribute the \( \frac{1}{2} \) exponent to each term inside the bracket: \( (x^2)^{1/2} \) and \( (y^8)^{1/2} \).
• For \( (x^2)^{1/2} \), apply the exponent rule: \( x^{2 \cdot (1/2)} \) which simplifies to \( x^1 \) or simply \( x \).
• Similarly, for \( (y^8)^{1/2} \), use the rule: \( y^{8 \cdot (1/2)} \) which simplifies to \( y^4 \).

By performing these actions carefully, you can convert a complex expression into a tidy one like \( x \cdot y^4 \), which uses rational exponents in its simplest form.

Positive Variables

In mathematics, assuming positive variables is important in problems involving roots and exponents. This assumption ensures that when you simplify expressions, values remain within real numbers and avoid complications like imaginary numbers.
When dealing with expressions such as \[ (x^2 y^8)^{1/2} \], assuming \( x \) and \( y \) are positive helps maintain the integrity of the solution. Here's why:

• If variables were negative and the exponents were fractional (like \( \frac{1}{2} \), which represents a square root), it could lead to non-real numbers.
• Positive variables ensure that when simplified to \( x \cdot y^4 \), the expression remains valid across real numbers without additional complexities.

This is why it’s often specified that all variables are positive in these types of problems—keeping everything streamlined and straightforward in the realm of real numbers.