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The concept of reciprocals is essential when dealing with negative exponents. A reciprocal is simply the flipped version of a fraction. In other words, if you start with a fraction like \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \). This flipping is quite handy because it helps to transform expressions involving negative exponents into something more manageable.

When you see a negative exponent, such as \( a^{-n} \), it's instructing you to first take the reciprocal of the base \( a \), and then apply the positive exponent \( n \). For example, with our given expression \( \left(\frac{3}{2}\right)^{-3} \), it turns into \( \frac{1}{\left(\frac{3}{2}\right)^3} \) using the reciprocal concept. This transformation simplifies solving the exercise because it converts a potentially complex problem into a multiplication task.

Multiplying fractions is a straightforward process, and it's crucial when raising a fraction to a power. When multiplying two fractions, you multiply their numerators together and their denominators together. For example, if you have fractions \( \frac{a}{b} \) and \( \frac{c}{d} \), their product is \( \frac{a \, \times \, c}{b \, \times\, d} \).

In our exercise, we need to compute \( \left(\frac{3}{2}\right)^3 \). This involves multiplying \( \frac{3}{2} \) by itself three times:

• Start by multiplying \( \frac{3}{2} \times \frac{3}{2} \), which gives \( \frac{9}{4} \).
• Then take \( \frac{9}{4} \) and multiply it again by \( \frac{3}{2} \), resulting in \( \frac{27}{8} \).

Notice how easy it is when you line up the fractions. Multiply across the numerators and across the denominators, and you arrive at the result with minimal confusion.

Raising a fraction to the power of three, or cubing it, means multiplying the fraction by itself twice more. When you see \( \left(\frac{a}{b}\right)^3 \), you should think of it as \( \frac{a}{b} \times \frac{a}{b} \times \frac{a}{b} \).

For the expression \( \left(\frac{3}{2}\right)^3 \), the arithmetic works like this:

• First multiplication: \( \frac{3}{2} \times \frac{3}{2} = \frac{9}{4} \).
• Second multiplication: Take the result \( \frac{9}{4} \) and multiply by \( \frac{3}{2} \), leading to \( \frac{27}{8} \).

Cubing a fraction keeps the operation simple and structured, as long as you remember to apply the multiplication process step-by-step. This repeated action may seem arduous, but it ensures accuracy and keeps errors at bay. From there, you can further manipulate the fraction as needed, such as finding its reciprocal if required.