Problem 10 - Algebra and Trigonometry

Question

\(7-28\) Evaluate each expression. $$ \left(2^{3}\right)^{0} $$

Step-by-Step Solution

Answer

The expression \((2^3)^0\) equals 1.

Step 1: Understanding the Expression

The expression given is \((2^3)^0\). It involves exponents, specifically an exponent raised to another exponent.

Step 2: Applying the Zero Exponent Rule

Any nonzero number raised to the power of zero is equal to one. Therefore, no matter what the base is, as long as it is nonzero, the entire expression \((2^3)^0\) simplifies to 1.

Key Concepts

Zero Exponent Rule

Understanding the zero exponent rule is crucial in simplifying expressions involving exponents. The rule states that any nonzero number raised to the power of zero equals one. This concept may seem counterintuitive at first, but here's why it works:

• Any number multiplied by itself any number of times keeps adding another factor of the number. When you multiply by zero times, you are left with one as there are no factors to consider.
• Mathematically, this helps keep patterns consistent across numbers and simplifies calculations. For example, consider the sequence of powers of 2: 2, 4, 8, 16, and so on. Reducing each result by a division by 2 (the base) until reaching the zero power results in one.

So, when you see \((2^{3})^{0}\), you can confidently simplify it to 1 without needing to calculate 2 raised to the third power first.

Evaluating Expressions

The task of evaluating expressions is all about breaking down complex expressions to find the simplest form or value. Here's how you can effectively evaluate an expression:

• Identify the parts of the expression: Determine the base, exponent, and operations involved.
• Apply rules and properties of exponents: Use applicable rules, such as the zero exponent rule or power of a power, to simplify the expression systematically.
• Simplify step by step: Follow a logical order of operations, often referred to as PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction).

For instance, when evaluating \((2^3)^0\), recognize that any power to zero simplifies to 1. Therefore, you can skip calculating \(2^3\) as it's unnecessary for this particular expression.

Base Exponentiation

Base exponentiation is the process of raising a base number to the power of an exponent. It fundamentally involves repeated multiplication of the base. Here’s what you need to remember:

• The base is the number being multiplied, and the exponent tells you how many times to use the base as a multiplier.
• For example, with \(2^3\), this means 2 multiplied by itself three times: 2 × 2 × 2 = 8.
• Understand that exponentiation grows numbers rapidly, which is why simplified rules like the zero exponent rule are so helpful when dealing with complex expressions.

In expressions such as \((2^3)^0\), although base exponentiation would typically involve calculating \(2^3\) first, the zero exponent rule immediately allows us to see that the entire expression simplifies to 1.