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In mathematics, the order of operations is crucial to solving expressions correctly. It dictates the sequence in which different mathematical operations should be performed. The typical order is: parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right). This is often remembered by the acronym PEMDAS.

In the expression \((2^3)^2\), we start with the innermost part of the expression, which is \(2^3\). According to the order of operations, exponents inside parentheses should be handled first. Once the inner exponentiation is resolved, we apply the next operation, which is the outer exponent \(^2\). Following this structured approach ensures that each operation is executed in its correct order, leading to the correct solution.

Simplification is the process of making a mathematical expression easier to understand or solve by reducing it to its simplest form. This often involves performing operations in the correct order to eliminate unnecessary complexity.

For the exercise \((2^3)^2\), simplification involves a two-step process:

• First, calculate the part of the expression inside the parentheses, \(2^3\), simplifying it to 8.
• Next, apply the outer exponent, raising the simplified result, 8, to the power of 2, which simplifies the expression to 64.

These steps transform the expression from its original form to a simpler numerical result.

The concept of powers, also known as exponents, is a way to define repeated multiplication of the same number. When we say \(2^3\), it means multiplying 2 by itself 3 times: \(2 \times 2 \times 2\). Here, 2 is the base, and 3 is the exponent.

In the expression \((2^3)^2\), we deal with powers raised to new powers. This can be approached by calculating the power inside the parentheses first, then applying the second power. According to the property of exponents \((a^m)^n = a^{m \cdot n}\), we could also directly compute it as \(2^{3 \times 2} = 2^6\), leading to the same result: 64. Understanding powers helps in recognizing patterns and properties that simplify calculations and deepen comprehension of exponential equations.