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The cube root of a number is essentially what number you would multiply by itself twice to achieve the original number. Let's think of it this way: if you have a number like 8, and you want to find its cube root, you are asking, "What number multiplied by itself three times gives me 8?" In this case, the answer is 2, since \(2 \times 2 \times 2 = 8\).

When dealing with larger numbers, like in our exercise \(\sqrt[3]{(-125)^2}\), the cube root can be tricky. It's important to first deal with any exponents like squaring or cubing, then proceed with finding the cube root. In the solution given, the cube root of 15625 is 25 because \(25 \times 25 \times 25 = 15625\).

This step-by-step approach helps in handling cube roots effectively, ensuring that each operation like squaring or finding the root is done one step at a time. It also helps to check your work at each stage for accuracy.

Exponential form is a method of expressing numbers where a base number is raised to a power or exponent. In our exercise, the given expression \((-125)^{2/3}\) is an example of a number written in exponential form where -125 is the base and 2/3 is the exponent.

Exponential notation is particularly useful because it allows you to simplify expressions and perform calculations more easily. For example, in this expression, the fraction \(\frac{2}{3}\) as an exponent shows us two operations: squaring the base of -125 and then taking the cube root.

Understanding how to manipulate exponential expressions involves recognizing that fractional exponents denote roots and powers. A fraction like \(\frac{1}{n}\) means taking the nth root, while a whole number indicates raising to a power. In this exercise, these principles combined to simplify and solve the problem effectively.

Simplification is the process of making a mathematical expression easier to work with or understand. When you simplify an expression, you aim to reduce it to its most basic form without altering its value.

For our example of \((-125)^{2/3}\), the simplification process involved several steps:

• Converting the expression into a root format by recognizing the fractional exponent \(\frac{2}{3}\) as two separate operations: squaring and taking the cube root.
• Performing operations in the correct order: first, computing \((-125)^2 = 15625\), then finding the cube root, which is 25.
• Revisiting results to ensure there are no computational errors in squaring or rooting steps.

Simplification not only makes the problem easier to solve, but also ensures that calculations are comprehensive and well-structured. Each operation must be verified to achieve an accurate result.