Problem 14 - Algebra for College Students

Question

Simplify each numerical expression. \(-\left(\frac{5}{6}\right)^{0}\)

Step-by-Step Solution

Answer

-1

Step 1: Understand the Problem

The expression given is \(-\left(\frac{5}{6}\right)^{0}\). To simplify this numerical expression, we need to follow the order of operations and evaluate any exponents within it.

Step 2: Evaluate the Exponent

The expression contains \(\left(\frac{5}{6}\right)^{0}\). According to the exponent rule, any non-zero number raised to the power of 0 is equal to 1. Thus, \(\left(\frac{5}{6}\right)^{0} = 1\).

Step 3: Apply the Negative Sign

The expression becomes \(-1\) after evaluating the exponent. The negative sign outside is a separate operation, so we simply apply it to the result of the exponent evaluation.

Step 4: Final Simplification

The expression \(-1\) is as simplified as possible, as there are no more operations to perform.

Key Concepts

Exponent Rules

Exponent rules are an essential part of simplifying expressions that involve powers. One of the basic rules to remember is that any non-zero number raised to the power of zero is always one. This simplifies complex expressions quickly by replacing any such power with one. For instance, consider the expression \((\frac{5}{6})^{0}\). The base, \(\frac{5}{6}\), does not matter in this particular rule as it's non-zero, which makes the expression equal to 1. This fundamental rule is helpful to easily evaluate expressions without complex calculations. Here are some quick pointers about the zero exponent rule:

• It's applicable to any non-zero base.
• It's a universal rule in mathematics, meaning it holds under any circumstances where the base is not zero.
• It simplifies expressions and equations, saving time and effort.

Order of Operations

The order of operations ensures that expressions are evaluated consistently and correctly. This is commonly remembered using the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).Applying PEMDAS to simplify expressions like \(-\left(\frac{5}{6}\right)^{0}\), you start by dealing with any parentheses first, which contains the exponent here. Evaluating the exponent takes precedence before applying multiplication or dealing with any prefix signs.Always remember when simplifying:

• Exponential evaluation comes immediately after resolving any operations in parentheses.
• Once the exponent is simplified, move to calculations outside the parentheses.
• Multiplicative negative signs are treated after exponent handling.

By following this structured approach, you maintain the correct simplification steps, ensuring accuracy.

Negative Numbers

When working with negative numbers, remember that a negative sign in front of an expression can significantly affect the final result. For our example, after evaluating the expression \((\frac{5}{6})^{0}\) to be 1, the negative sign in \(-1\) must be considered next. In mathematics, the negative sign denotes the opposite of a number. So, if your expression yields \(1\), then with the negative sign, it becomes \(-1\). Key points to understand about negative numbers:

• They represent values less than zero.
• Positioning changes the expression's outcome, shifting it lower on the number line.
• They are often represented in a distinct manner to denote their difference from positive counterparts.

Seeing how negative signs influence outcomes helps in accurately evaluating entire expressions.