Problem 8 - Beginning and Intermediate Algebra

Question

Simplify. $$\left(-\frac{2}{5}\right)^{0}$$

Step-by-Step Solution

Answer

The simplified form of the expression \(\left(-\frac{2}{5}\right)^{0}\) is 1.

Step 1: Apply the exponent rule for raising a non-zero number to the power of 0

According to the exponent rule, any non-zero number raised to the power of 0 is equal to 1. Therefore, we can apply this rule to the given expression: \[\left(-\frac{2}{5}\right)^{0} = 1\]

Step 2: Write the final simplified expression

The expression has been simplified, and there are no more steps to perform: \[\left(-\frac{2}{5}\right)^{0} = 1\] The simplified form of the given expression is 1.

Key Concepts

Zero Exponent Rule

The zero exponent rule is a fundamental concept in algebra that states any non-zero number raised to the power of zero equals one. This rule might seem a bit strange at first, but it's especially handy when you encounter expressions involving exponents. Imagine an expression such as \(a^0\). According to the zero exponent rule, as long as \(a\) is not zero, \(a^0 = 1\).
For instance, let's consider the expression \(\left(-\frac{2}{5}\right)^0\). Here, the base is a fraction, \(-\frac{2}{5}\). Despite being a fraction and a negative number, as long as it isn't zero, the rule applies. Thus, \(\left(-\frac{2}{5}\right)^0 = 1\).
This rule is crucial as it simplifies calculations and mathematical expressions by reducing complex equations to something manageable. Remember, the rule does not apply if the base is zero. For \(0^0\), the expression is indeterminate and context-dependent.

Simplifying Expressions

Simplifying expressions in algebra is the process of reducing complexity while maintaining the expression's original meaning. The aim is to make the expression easier to read, understand, and work with. This often involves using rules and properties of numbers and operations, such as the zero exponent rule.
When simplifying an expression like \(\left(-\frac{2}{5}\right)^0\), we applied the zero exponent rule, reducing the fraction to the number 1. This one-step simplification is powerful because it saves time and prevents errors in larger problems.
To simplify effectively:

• Identify and apply exponent rules where relevant, like the zero or negative exponent rules.
• Combine like terms, using basic arithmetic operations.
• Always check each step, ensuring no details are overlooked.

These strategies streamline calculations and provide clarity, especially when dealing with intricate algebraic equations.

Algebra Basics

Algebra basics cover the fundamental concepts and rules that form the foundation of algebra, providing essential tools for solving equations and understanding mathematical relationships. It's crucial for students beginning their exploration of math.
At its core, algebra deals with finding the unknown values represented by variables, performing operations with these variables, and solving equations. Familiarizing yourself with the core rules, like exponent rules, is critical. For example:

• The power of zero rule states \(x^0 = 1\) for any \(x eq 0\).
• When multiplying powers with the same base, add the exponents: \(x^a \cdot x^b = x^{a+b}\).
• For dividing powers with the same base, subtract the exponents: \(\frac{x^a}{x^b} = x^{a-b}\).

These rules simplify tasks such as solving algebraic problems, working with polynomials, and graphing equations. Mastery of algebra basics, including understanding how and when to apply these exponent rules, paves the way for advancing in mathematics and other scientific fields.