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Exponentiation is a fundamental operation in mathematics that involves raising a number, called the base, to the power of an exponent. The exponent tells us how many times to multiply the base by itself. For example, in the expression \(3^4\), 3 is the base, and 4 is the exponent, meaning we multiply 3 by itself 4 times: \(3 \times 3 \times 3 \times 3 = 81\).

When dealing with fractional exponents, the operation involves both roots and powers. A fractional exponent like \(a^{m/n}\) means you first take the n-th root of a, then raise the result to the m-th power. This dual process may seem complex, but it's manageable when you break it down into these simple steps.

The cube root of a number is a special value that, when multiplied by itself three times, equals the original number. For example, the cube root of 64 is 4 because \(4 \times 4 \times 4 = 64\). You may see the cube root written as \(64^{1/3}\), which means the same thing.

In the context of our exercise, finding the cube root is a critical step. First, we rewrite the expression \(64^{2/3}\) as \((64^{1/3})^2\). By doing this, we simplify the problem: we first find the cube root of 64 (which is 4), and then square the result (because of the 2 in the exponent).
\(64^{1/3} = 4\)
Next, we find \((4)^2 = 16\). Thus, \(64^{2/3} = 16\).

The power rule is another fundamental concept in exponentiation that states \((a^m)^n = a^{m \times n}\). This rule allows us to simplify expressions involving powers of powers. In our exercise, we rewrite \((64^{2/3})\) as \((64^{1/3})^2\).

This step translates the problem into finding the cube root first, then raising that result to the second power. By using the power rule, we make the computation more manageable:
- Start with \(64^{1/3}\) to get the cube root of 64, which is 4.
- Then, \( (4)^2 = 16\).
Therefore, \(64^{2/3} = 16\).
These steps utilize the power rule to simplify the complex expression.

Simplification is the process of breaking down a mathematical problem into more manageable parts, making it easier to solve. This concept is vital in solving expressions with fractional exponents.

Let's look at our exercise again. The original expression is \64^{2/3}\. Using simplification, we first break it down by rewriting it as \( (64^{1/3})^2 \). This step involves translating a fractional exponent into a root and a power, handled separately:
- Finding \( 64^{1/3} \), the cube root of 64, gives us 4.
- Then, we solve \( (4)^2 \), squaring our intermediate result to get 16.

By simplifying the problem step-by-step, we make sure our calculations are accurate and understandable. Therefore, \( 64^{2/3} = 16 \). Simplification is all about making complex problems easier to work with!