Problem 37 - Intermediate Algebra

Question

Simplify each expression. Assume that the variables can be any real number, and use absolute value symbols See Example 2. $$ \left(-64 x^{8}\right)^{1 / 8} $$

Step-by-Step Solution

Answer

The simplified expression is \(|x|\).

Step 1: Understand the Expression

The expression to simplify is \((-64 x^{8})^{1/8}\). This is a power expression involving a negative base and a variable raised to a power inside the base. We will simplify this by dealing with the base and the exponent separately.

Step 2: Simplify the Numeric Base

First, consider the numeric base of the expression which is \(-64\). The eighth root of a number \(a\) is \(a^{1/8}\). Because \(-64\) is negative and we're taking an even root of a real number, we need to express the solution in terms of absolute value. The eighth root of \(-64\) is not real because negative numbers do not have even roots in the real number system without using complex numbers. We express it as \((-1)^{1/8} \times 64^{1/8}\).

Step 3: Simplify the Exponent on the Variable

Next, look at the variable part of the expression \(x^8\). The rule for exponents is \((a^m)^n = a^{m\times n}\). Here, \((x^8)^{1/8} = x^{8/8} = x^1 = x\). Hence, \(x\) remains unchanged after simplification.

Step 4: Combine the Simplified Parts

Combine the simplified parts: We have \((-1)^{1/8} \times 64^{1/8} \times x\). Since \(64^{1/8}\) equals \(64^{1/8} = 2^{(6/8)} = 2^{3/4}\). Expressing the complex number separately, the solution in terms of its absolute value becomes \(|x|\), as \(x^8\) raised to the one-eighth power is the absolute value of \(x\).

Step 5: Rewrite Using Absolute Value

Even roots of real numbers expressed negatively should consider absolute value to indicate real solution relationships. Therefore, \(((-64 x^{8})^{1/8} = x)\), capturing real number solutions. The absolute value function applies because raising base \(x\) under even exponentiation contraction universally guarantees non-negativity when roots are extracted. In conclusion, write \(\left| x \right|\).

Key Concepts

Simplifying Expressions

When we talk about simplifying expressions, it's like finding the shortest route to an answer. We aim to take a complex mathematical expression and break it down into something more manageable, without changing its value. This involves understanding the different parts of the expression, including numbers, variables, and operations.

Simplifying expressions often requires us to:

• Identify like terms, such as terms with the same variables raised to the same powers, and combine them.
• Apply mathematical operations correctly, such as addition, subtraction, and division.
• Follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

The ultimate goal is to express the equation as simply as possible, leading us to uncover the easiest-to-read version of the mathematical truth it holds.

In our original exercise, simplifying involved both numeric and variable terms. We processed the negative base separately from the variable part, simplifying each before combining them. Each step required an assessment of whether further simplification was possible.

Absolute Value

The concept of absolute value might seem a little tricky at first, but think of it as the distance a number is from zero on a number line. This means that absolute values are always non-negative, no matter if they're derived from positive or negative numbers.

Here's a quick way to think about it:

• The absolute value of a positive number is the number itself (e.g., \( |5| = 5 \)).
• The absolute value of a negative number is its positive counterpart (e.g., \( |-5| = 5 \)).

This is particularly crucial when dealing with even roots like square roots, fourth roots, etc., where we cannot directly take the root of a negative number and remain within the real number system. That's why in our problem, \( x^8 \) raised to the \( 1/8 \) power was simplified to \( |x| \). This way, we ensure that the result is always a real, non-negative figure, capturing the essence of the distance from zero rather than the directional value of the original expression.

In mathematics, ensuring that results make sense within the real number framework sometimes requires taking absolute value into account, especially when even roots are involved.

Exponents and Roots

Exponents and roots are interconnected concepts in mathematics that allow us to express repeated multiplication and the inverse of those operations, respectively. Exponents indicate how many times we multiply a number by itself, while roots demonstrate how many times a number has been multiplied to reach its current form.

Some important points to remember:

• The expression \( a^m \) means to multiply \( a \) by itself \( m \) times.
• The root, written as \( a^{1/n} \), represents the inverse operation of taking an \( n \)-th root of a number.
• When working with exponents and roots, rules such as \( (a^m)^n = a^{m\cdot n} \) and \( a^{m/n} = \sqrt[n]{a^m} \) are fundamental tools.

In our original problem, we encountered the expression \( (-64 x^8)^{1/8} \). Handling this required separate consideration of numeric and variable components with their respective roots. The numeric part, \( -64 \), needed special treatment due to its negativity when dealing with an even root, leading us into potential complex number territory. By using absolute values with the variable part \( x^8 \), we isolated the real numbers retained through root simplification, keeping our final expression clean and conforming to real number expectations.