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Fractions are made up of two main parts: the numerator and the denominator.
The numerator is the number on the top of the fraction. It tells us how many parts we are considering.
The denominator is the number on the bottom of the fraction. It shows the total number of equal parts the whole is divided into.
For example, in the fraction \(\frac{3}{4}\), 3 is the numerator and 4 is the denominator.
Understanding these two parts is essential because fraction operations often need us to manipulate each part separately.
For instance, when simplifying or comparing fractions, we might change the numerator while keeping the denominator the same.

Exponents are a way to express repeated multiplication of the same number.
In mathematical notation, \(a^n\) means multiplying the number \(a\) by itself \(n\) times.
For example, \(3^3\) means \(3 \times 3 \times 3 = 27\).
When working with exponents in fractions, you apply the exponent to both the numerator and the denominator if the entire fraction is raised to a power.
For instance, \(\left( \frac{3}{4} \right)^2\) means raising both 3 and 4 to the power of 2. This results in \(\frac{3^2}{4^2} = \frac{9}{16}\).
Knowing how to handle exponents is crucial for simplifying complex fractions in exercises.

Manipulating fractions involves a set of operations like addition, subtraction, multiplication, and division.
Simplifying fractions means reducing them to their simplest form, where the numerator and the denominator share no common factors other than 1.
To add or subtract fractions, make sure they have the same denominator. If not, find a common denominator first.
Multiplying fractions is straightforward: simply multiply the numerators together and the denominators together.
\(\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}\)
For division, multiply by the reciprocal of the second fraction: \(\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}\)
For example, in the exercise, combining and simplifying the fractions led us to solve \(\frac{18 \times 16}{9} = \frac{288}{9} = 32\).