Problem 22 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. $$ (-5)^{0} $$

Step-by-Step Solution

Answer

1

Step 1: Determining the base and the exponent

In the given expression \((-5)^{0}\), -5 is the base and 0 is the exponent.

Step 2: Applying the power of zero rule

By the rule of exponents stating that any number (except zero) raised to the power of 0 is 1, the expression \((-5)^{0}\) evaluates to 1.

Key Concepts

Power of zero

When dealing with exponents, the power of zero presents an interesting rule. In mathematics, any non-zero number raised to the power of zero is always equal to 1. This might seem confusing at first, because we're used to the fact that multiplying by zero results in zero. However, the idea behind the power of zero is different. It's about how many times the base is multiplied by itself.

Let's simplify this idea:

• If you have \(5^3\), this means \(5\) multiplied by itself three times (\(5 \times 5 \times 5\)).
• For \(5^1\), it means the number 5 only once.
• But what about \(5^0\)? It means the number 5 doesn't multiply at all, which mathematically results in 1.

The rule of zero power is foundational in exponentiation and applies to any base except zero itself, as \(0^0\) can be ambiguous and isn’t typically defined in basic algebra contexts.

Base and exponent

In exponentiation, understanding the terms 'base' and 'exponent' is crucial. These terms define the structure of something called a power. The base is the number that is being multiplied, whereas the exponent indicates how many times the base is multiplied by itself. Here's how it works:

• The base is the main number you see in the expression. For example, in \(x^y\), 'x' is the base.
• The exponent is a small written number, or power, to the upper right of the base. It tells you how many times to multiply the base by itself. So, in \(x^y\), 'y' is the exponent.
• If the exponent is 0, as in \((-5)^0\), the rules of exponents described earlier come into play.

It's like a shorthand in mathematics to express repeated multiplication. The expression \(2^3\) tells us to multiply the base, 2, by itself three times: \(2 \times 2 \times 2 = 8\). This simplification makes it easier to handle large numbers and complex calculations.

Rules of exponents

The rules of exponents are vital guidelines in mathematics that simplify calculations involving powers. Here are some key rules to help you:

• Product of Powers Rule: When multiplying two powers with the same base, add the exponents: \(a^m \times a^n = a^{m+n}\).
• Quotient of Powers Rule: When dividing two powers with the same base, subtract the exponents: \(a^m / a^n = a^{m-n}\).
• Power of a Power Rule: When you have a power raised to another power, multiply the exponents: \((a^m)^n = a^{m \cdot n}\).
• Zero Exponent Rule: Any non-zero base raised to the zero power is equal to 1: \(a^0 = 1\).

Understanding these rules allows you to solve exponent-related problems with ease. Each rule helps reduce complex expressions to more manageable forms, making mathematics less daunting and more systematic. Apply these rules whenever you're working with powers and you'll find many problems become simpler to tackle.