Problem 40 - Intermediate Algebra

Question

Simplify each expression. Assume that the variables can be any real number, and use absolute value symbols See Example 2. $$ \left[(x+1)^{6}\right]^{1 / 6} $$

Step-by-Step Solution

Answer

The expression simplifies to \( x+1 \).

Step 1: Understand the Expression

Look at the expression \( [(x+1)^6]^{1/6} \) and notice that it involves an exponent raised to another exponent. This is a situation where the power rule of exponents can be used: \( (a^m)^n = a^{m imes n} \).

Step 2: Apply the Power Rule of Exponents

Apply the power rule \( (a^m)^n = a^{m imes n} \) to simplify \( [(x+1)^6]^{1/6} \). So we calculate \( 6 \times \frac{1}{6} \), which equals 1. This simplifies our expression to \( (x+1)^1 \).

Step 3: Simplify the Expression

Since any expression to the power of 1 is itself, \( (x+1)^1 \) simplifies to \( x+1 \). So the simplified expression is \( x+1 \).

Key Concepts

Simplifying Expressions

In mathematics, simplifying expressions is a fundamental skill that allows us to make complex problems more manageable. When we simplify an expression, we aim to express it in its simplest form, removing any redundancies or complications.

Consider the expression \([(x+1)^6]^{1/6}\). The goal here is to simplify it by removing the nested exponents. We employ the power rule of exponents, which states that \((a^m)^n = a^{m \times n}\). By applying this rule, we can reduce the expression to a simpler form.
- First, multiply the exponents: \(6 \times \frac{1}{6} = 1\).
- This leaves us with \((x+1)^1\), which simply equals \(x+1\).

Simplifying expressions like these helps us solve and understand equations more easily. It's an essential step in many areas of algebra and beyond.

Real Numbers

The concept of real numbers is pivotal in mathematics, as they encompass almost all the numbers you'll encounter in calculations. Real numbers include rational numbers, like 3, -5, and 7.2, and irrational numbers, like \(\pi\) and the square root of 2.

- Rational numbers can be expressed as a ratio of two integers, such as \(\frac{7}{3}\), whereas irrational numbers cannot.

In our exercise, you are told that the variables can be any real number. This means the variable \(x\) in \([(x+1)^6]^{1/6}\) can take any value along the number line. Understanding this sets the stage for manipulating and simplifying expressions, as it identifies the domain of possible values for \(x\).

Real numbers are crucial because they offer a complete picture of the solutions that an equation or expression can have.

Absolute Value

Absolute value is a concept that refers to the distance of a number from zero on the number line, regardless of direction. It is always expressed as a non-negative value.

For instance, the absolute value of both -5 and 5 is 5. The absolute value simply measures how far the number is from zero.

Although our specific exercise doesn't require extensive manipulation of absolute values, it's important to note them when simplifying expressions.

• They ensure clarity while dealing with variables that can represent any real number.
• Absolute value signs are crucial when dealing with solutions to equations that may involve negative results, ensuring we represent only the magnitude of these answers.

In algebra, keeping the idea of absolute value in mind allows us to work with expressions more flexibly and accurately. This amplifies the reliability of our simplifications in real-world applications.