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Multiplying fractions is a straightforward process. In essence, you're working with two main parts: the numerators (the top parts of the fractions) and the denominators (the bottom parts). When you multiply two fractions, you multiply their numerators to get a new numerator, and their denominators to get a new denominator. For example, if you're looking at two fractions \(\frac{a}{b}\) and \(\frac{c}{d}\), to multiply them, you do:

• Numerator of the result: \(a \times c\)
• Denominator of the result: \(b \times d\)

So, in the example given in the exercise, multiplying \(\frac{7}{10}\) by \(\frac{5}{2}\) results in \(\frac{35}{20}\) by performing the operations:

• \(7 \times 5 = 35\)
• \(10 \times 2 = 20\)

Exponents represent repeated multiplication of a number by itself. When you multiply terms that have the same base, you apply the exponent rules. Specifically, you add the exponents together. This rule stems from the fundamental idea that exponents denote how many times a number is used in a multiplication.
For example, \(d^9\times d^2\) involves multiplying \(d\) to itself nine times and two more times, resulting in \(d^{9+2}\) or \(d^{11}\). This shows how powerful the base becomes when combining exponents.
This specific rule applies only to terms with the same base, such as \(d\) in this exercise. Each base handles its own exponents separately unless shared by other operations. By properly adding the exponents, you can simplify complex expressions effectively while maintaining precision.

After you multiply fractions, it's important to check if you can simplify the resulting expression. Simplifying fractions means reducing them to their simplest form, where the numerator and the denominator have no common factors other than 1.
To simplify a fraction, you look for the greatest common divisor (GCD) of the numerator and the denominator. Divide both terms by this number to find the simplest form. For instance, with \(\frac{35}{20}\), the GCD is 5. Dividing both the numerator (35) and the denominator (20) by 5 results in \(\frac{7}{4}\).

• 35 divided by 5 is 7.
• 20 divided by 5 is 4.

Simplifying helps express the fraction more compactly and makes further calculations easier. Moreover, it leads to cleaner results that are easier to interpret and manipulate in subsequent algebraic work.