Problem 29 - Intermediate Algebra

Question

Simplify each expression. Assume that the variables can be any real number, and use absolute value symbols See Example 2. $$ \left(4 x^{4}\right)^{1 / 2} $$

Step-by-Step Solution

Answer

The simplified expression is \( 2x^{2} \).

Step 1: Understand the Expression

The given expression is \( \left(4 x^{4}\right)^{1 / 2} \). Our goal is to simplify this expression by applying the rules of exponents.

Step 2: Apply the Power of a Product Rule

The Power of a Product Rule states that \( (ab)^{m} = a^{m}b^{m} \). Apply this rule to the expression: \( \left(4 x^{4}\right)^{1 / 2} = 4^{1/2} \cdot (x^{4})^{1/2} \).

Step 3: Simplify Each Part Separately

Simplify the constant part: \( 4^{1/2} = \sqrt{4} = 2 \). Then, simplify the variable part: \( (x^{4})^{1/2} = x^{4 \cdot (1/2)} = x^{2} \).

Step 4: Combine the Results

Now combine the simplified constant and variable terms: \( 2 \cdot x^{2} \), which becomes \( 2x^{2} \). Since the expression inside the radical involves an even power of \( x \), we usually need absolute value bars, but since \( x^2 \) is non-negative, absolute values are not necessary in this context.

Key Concepts

Exponent Rules

Exponent rules are essential in simplifying expressions involving powers. When working with exponents, the goal is to understand how they interact with each other and with the numbers and variables they are applied to. Here are some basic exponent rules to keep in mind:

• Product of Powers Rule: This rule is used when you multiply two expressions that have the same base. It says that you add the exponents: \( a^{m} \cdot a^{n} = a^{m+n} \).
• Power of a Power Rule: This rule is used when raising an exponent to another power. It states that you multiply the exponents: \((a^{m})^{n} = a^{m\cdot n} \).
• Power of a Product Rule: This rule is crucial when you have a product inside a set of parentheses raised to a power: \((ab)^{m} = a^{m}b^{m}\).
  
• Zero Exponent Rule: Any base (except zero) raised to the zero power is equal to one: \( a^{0} = 1 \).
• Negative Exponent Rule: This tells you to take the reciprocal of the base: \( a^{-m} = \frac{1}{a^{m}} \).

Understanding these rules allows you to manipulate and simplify expressions effectively, making them easier to work with.

Power of a Product Rule

The "Power of a Product Rule" is extremely useful when you need to simplify expressions like \((ab)^{m}\). It allows you to distribute the exponent to both parts of the product separately. Here's how it works in detail:

Suppose you have an expression involving a product raised to a power, such as \((4x^{4})^{1/2}\). According to the rule, you can separately apply the power to each component of the product: \((4)^{1/2}\times(x^{4})^{1/2}\).

• This separation makes it easier to simplify complex expressions step by step.
  
• For the constant part: \((4)^{1/2} = 2\) because 4 is a perfect square.
• For the variable part: \((x^{4})^{1/2} = x^{2}\). This uses the Power of a Power Rule as \(4\cdot\frac{1}{2} = 2\).
  

Combining the results from simplifying each part gives you \(2 \cdot x^{2} = 2x^{2}\). This approach, by dealing with each part step by step, makes simplification much more straightforward and manageable.

Absolute Value in Algebra

In algebra, the absolute value of a number is its distance from zero on the number line, without considering direction. It is always non-negative. Mathematically, it's denoted as \(|x|\). Here's how you can understand and apply it:

• Multiple Forms:
  Absolute value affects both positive and negative numbers. For instance, \(|3| = 3\) and \(|-3| = 3\). Both are 3 units away from zero.
  
• Utilizing Absolute Value:
  In expressions, especially those involving variables, absolute value ensures that the expression remains non-negative. When simplifying expressions like \(x^{2}\), absolute value isn't needed because the result is always non-negative. However, for an expression like \(x\), to keep it in absolute value form, you might write it as \(|x|\) if it could potentially be negative.
  
• When to Skip Absolute Value:
  In the context of the exercise \((4x^{4})^{1/2} = 2x^{2}\), absolute values are not necessary. This is because \(x^{2}\) is inherently non-negative, thereby making \(|x^{2}|\) the same as \(x^{2}\).

Understanding where and when to apply absolute values helps ensure expressions are presented accurately and in the simplest form.