Problem 63 - Algebra A Combined Function

Question

Simplify each expression. $$ 7^{0} $$

Step-by-Step Solution

Answer

7⁰ simplifies to 1.

Step 1: Understand the Zero Exponent Rule

According to the zero exponent rule, any nonzero number raised to the power of zero is equal to one. This is a fundamental rule in exponents and applies to our expression 7⁰.

Step 2: Apply the Zero Exponent Rule

Apply the zero exponent rule to the expression 7⁰. Since 7 is a nonzero number, 7 raised to the power of zero is: \[ 7^0 = 1 \]

Key Concepts

Understanding Exponents

Exponents are a way to express repeated multiplication of a number by itself. The base number is the number being multiplied, and the exponent tells us how many times the base number is used as a factor in the multiplication. For example, in the expression \(3^4\), 3 is the base and 4 is the exponent. This means that 3 is multiplied by itself four times, or \(3 \times 3 \times 3 \times 3 = 81\).

Exponents are a powerful tool in mathematics because they allow us to write large numbers or repeated operations in a compact form. They are used in many areas such as algebra, calculus, and beyond, making them a crucial concept to understand.

Simplifying Expressions

Simplifying mathematical expressions involves reducing them to their simplest form, making them easier to understand or compute. When dealing with expressions involving exponents, simplification might include applying various exponent rules, such as the zero exponent rule.

To simplify an expression like \(7^0\), we apply the zero exponent rule, which states that any nonzero base raised to the power of zero equals one. Thus, \(7^0\) simplifies to 1. This practice helps eliminate complexity in expressions and focuses on their essential elements, a key skill in mathematics.

Mathematical Rules in Simplification

Mathematical rules, such as the zero exponent rule, play a significant role in simplifying expressions. The zero exponent rule is simple but mighty, as it applies broadly: if you have a base of any nonzero number, and it's raised to an exponent of zero, the result is always one. This not only simplifies the specific case of exponents but also exemplifies the broader structure and consistency found in mathematics.

Rules like these help us navigate complex expressions effortlessly and are foundational in algebra and beyond. Remember :

• Zero exponent rule: \(a^0 = 1\), for any nonzero \(a\).
• Understanding these rules will provide you with the tools to simplify more intricate mathematical expressions.