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When we talk about evaluating expressions, we essentially mean finding the value of a mathematical phrase. These phrases can involve numbers, variables, and mathematical operators like addition, subtraction, multiplication, and division. Evaluating requires understanding the order of operations, commonly remembered by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division, Addition and Subtraction.

In the original exercise, the expression to evaluate is \(3 \cdot 2^{6}\). Following PEMDAS, we first need to deal with the exponentiation before moving on to multiplication. Evaluating accurately involves breaking down these steps into manageable parts, as it helps simplify the problem and ensure correctness.

Multiplication is a mathematical operation that combines equal groups. It is one of the basic arithmetic operations and can be viewed as repeated addition. In the expression \(3 \cdot 64\), multiplication is used to scale up the value obtained from evaluating the power.

Consider that \(3 \cdot 64\) can be thought of as adding the number 64 three times:

• 64 + 64 + 64

The exercise showcases the second step of solving the original problem, where the result of the exponent \(64\) is multiplied by the given coefficient \(3\). This process results in the final evaluated value. Multiplication helps in quickly finding the sum of equal numbers, rather than adding them one by one.

The concept of exponentiation or the power of a number involves raising a base number to an exponent. This indicates how many times the base number needs to be multiplied by itself. For instance, in the expression \(2^{6}\), 2 is the base and 6 is the exponent, representing \(2\) multiplied by itself 6 times.

Let's break it down into steps:

• Start with 2
• Multiply by 2: \(2 \times 2 = 4\)
• Multiply again by 2: \(4 \times 2 = 8\)
• Continue until it's done 6 times: \(8 \times 2 = 16\), \(16 \times 2 = 32\), \(32 \times 2 = 64\)

Through these calculations, \(2^{6}\) equals 64. Understanding the "power of a number" is essential since it simplifies expressing and computing large numbers in practical problems.