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Rational exponents are a way to express powers and roots in mathematical expressions. Unlike a simple integer exponent, rational exponents involve fractions. When an exponent is a fraction, such as in the expression \(a^{m/n}\), it means two things:

• Raise the base \(a\) to the power of the numerator \(m\).
• Take the root indicated by the denominator \(n\).

This is expressed as: \[ a^{m/n} = \sqrt[n]{a^m} \] This means you first perform the power operation, and then the root operation. Alternatively, you can also perform the root operation first and then the power operation, i.e., \( \sqrt[n]{a}^m \). The order doesn't affect the final result. Understanding rational exponents is crucial when simplifying expressions involving roots and powers. It streamlines calculations by combining the power and root into a single expression.

The cube root is a specific type of root operation that involves finding a number which, when multiplied by itself three times, gives the original number. The notation for the cube root of a number \(x\) is \(\sqrt[3]{x}\), and it fulfills the equation \(y^3 = x\), where \(y\) is the cube root.Cube roots are useful when dealing with expressions that involve dimensional measurements in geometry or evaluating mathematical expressions in algebra. Unlike square roots, cube roots apply not only to perfect cubes (like \(8\), \(27\), etc.), but also to negative numbers. This is because a negative number multiplied by itself three times results in a negative product.For instance, the cube root of \(-27\) would be \(-3\) because \((-3) \times (-3) \times (-3) = -27\).Applying this to our example, taking the cube root of \(64\) means finding a number that, when cubed, equals \(64\). As demonstrated in the step-by-step solution, this number is \(4\) because \(4 \times 4 \times 4 = 64\).

Evaluating expressions typically involves simplifying them to obtain a final, simpler form. This involves understanding and applying arithmetic operations, exponents, roots, and any applicable rules or properties. Let’s walk through evaluating an expression like \(8^{2/3}\) based on the steps given:1. **Power Operation**: This step involves taking the base \(8\) and raising it to the power indicated by the numerator (\(2\)), resulting in \(64\). This is simply multiplying \(8\) by itself: \(8 \times 8 = 64\).2. **Root Operation**: After obtaining \(64\) from the power operation, the next task is to find the cube root of \(64\), as indicated by the denominator (\(3\)). This step reverses cubing and finds a number that, when multiplied three times, equals \(64\). Thus, we find that \(4\) is the cube root of \(64\).By breaking it down, you ensure accurate computation without a calculator. Always remember:

• Follow the order of operations: exponents before roots can vary unless specified.
• For increased understanding, try solving manually and verify with technology where necessary.