Problem 18 - Algebra 2 and Trigonometry

Question

In \(11-22,\) find the value of each expression when \(x \neq 0\) $$ (-2 x)^{0} $$

Step-by-Step Solution

Answer

The value of the expression \((-2x)^{0}\) is 1.

Step 1: Understand the Expression

In the expression \((-2x)^{0}\), we have a term raised to the power of zero. Any non-zero number raised to the power of zero equals 1.

Step 2: Apply the Zero Exponent Rule

Using the rule that any non-zero base raised to the power of zero is 1, calculate \((-2x)^{0}\). Since \(-2x\) is a non-zero expression, it follows that \((-2x)^{0} = 1\).

Key Concepts

Exponents

Exponents are a fundamental concept in mathematics, representing repeated multiplication. When you see a number with an exponent, like \(a^b\), it means you multiply \(a\) by itself \(b\) times. For example, \(3^2\) means \(3 \times 3\), which equals 9. This is a simple way of expressing large multiplications in a compact format. Exponents can be positive, negative, or even zero, each holding its own unique rules and applications.

In the context of the zero exponent rule, understanding that exponents can be anything other than positive integers expands our grasp on how flexible and powerful this notation can be. Familiarity with these rules can enhance your ability to solve complex mathematical problems with ease.

• Exponents show how many times a number, the base, is multiplied by itself.
• They simplify expressions and calculations.
• Exponents can be applied to both numbers and algebraic expressions.

Non-zero Base

A non-zero base refers to any value or expression that is not equal to zero. It's important because many mathematical operations, like division and exponentiation, require a non-zero base to avoid undefined results.

In the expression \((-2x)^{0}\), the term \(-2x\) is considered the base. Despite it containing the variable \(x\), as long as \(x eq 0\), the base remains non-zero.

• Ensure the base value is not zero to correctly apply the zero exponent rule.
• The base can be a number or any mathematical expression that simplifies to a non-zero value.
• Expressions with variables are common, requiring careful substitution to check for non-zero conditions.

Power of Zero

The power of zero, often termed the zero exponent rule, is a critical mathematical shortcut that significantly simplifies calculations. This rule states that any non-zero number or expression raised to the zero power equals 1. So, if you encounter \((a)^{0}\), the result is 1 provided that \(a\) is not zero, confirming the rule's applicability.

The reason this rule works stems from the properties of exponents themselves. If you think about decreasing exponents, moving from \(a^n\) to \(a^{n-1}\), each step represents dividing by the base \(a\). Extending this logic, reaching zero leads to dividing by the base n times, yet the value must remain consistent with its identity, resulting in 1.

• Any non-zero expression involving variables, like in our example, follows this rule strictly.
• Remembering the power of zero rule can save time and effort in various calculations.
• Using \((-2x)^{0} = 1\) is an example of applying this handy shortcut.