Problem 6 - Algebra 1

Question

Evaluate the expression. \(6 \cdot 3^{0}\)

Step-by-Step Solution

Answer

The solution to the expression \(6 \cdot 3^{0}\) is 6.

Step 1: Identify exponent expression

The expression involves a multiplication and an exponentiation. Here, the exponent expression is \(3^{0}\).

Step 2: Evaluate exponent expression

In mathematics, any non-zero number raised by zero equals one. Hence, \(3^{0} = 1\).

Step 3: Substitute and Simplify

Substitute \(3^{0}\) = 1 into the original expression: \(6 \cdot 1 = 6\).

Key Concepts

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, known as the base, to the power of another number, called the exponent. The base is multiplied by itself as many times as indicated by the exponent. For example, in the expression \(b^n\), \(b\) is the base, and \(n\) is the exponent. This operation is built on the idea of repeated multiplication.
It's important to understand the basic behavior of exponentiation:

• If the exponent is 1, the result is the base itself, e.g., \(b^1 = b\).
• An exponent of 0 has a special rule, which we'll explore in the zero exponent rule section.
• Negative exponents indicate a reciprocal, e.g., \(b^{-n} = \frac{1}{b^n}\).

In our exercise, the expression \(3^0\) shows an example of an exponentiation that follows a specific rule given the exponent's value.

Multiplication

Multiplication is one of the basic arithmetic operations, where you combine equal groups of numbers. It can be understood as repeated addition. For example, multiplying 4 by 3 is the same as adding 4 three times: \(4 + 4 + 4 = 12\). This is expressed as \(4 \times 3 = 12\).
In the context of our exercise, we encounter the expression \(6 \cdot 3^{0}\). Here, multiplication applies after we simplify the exponential part of the expression. Once we know that \(3^0 = 1\), we multiply 6 by 1, which maintains the value of 6, as multiplying any number by 1 leaves it unchanged.
This demonstrates how multiplication interacts with other operations like exponentiation to simplify expressions.

Zero Exponent Rule

The zero exponent rule is a fascinating and handy guideline in mathematics. It states that any non-zero number raised to the power of zero is equal to one. Symbolically, for any number \(a\), \(a^0 = 1\) provided \(a eq 0\).
Understanding why this works involves considering the properties of exponents. When dividing powers with the same base, you subtract the exponents: \(b^m / b^m = b^{m-m} = b^0 = 1\). Despite dividing the same quantity, the result equals 1, not zero, which justifies the rule.
In our exercised expression \(3^{0}\), the rule simplifies the calculation significantly, reducing \(3^0\) directly to 1. This simplification emphasizes exponentiation's power in making complex calculations more manageable.