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Understanding fraction multiplication is crucial for solving problems involving division of fractions. When multiplying fractions, you multiply the numerators together and the denominators together. For instance, if you want to multiply \(\frac{a}{b} \times \frac{c}{d}\), you simply do the following steps:

• Multiply the numerators: \a \times c\
• Multiply the denominators: \b \times d\

. The result will be \(\frac{a \times c}{b \times d}\). This method can be directly applied when you convert a division problem into multiplication by using the reciprocal of the divisor. Remember to always multiply the numerators and then the denominators.

For example, in the problem \(-\frac{4}{3} \times -8\), you do:

• Numerator: \-4 \times -8 = 32\
• Denominator: \3 \times 1 = 3\

.
Thus, you get \(\frac{32}{3}\).

Understanding the concept of the reciprocal is essential for dividing fractions. The reciprocal of a fraction \(\frac{a}{b}\) is simply \(\frac{b}{a}\). If you have a whole number, you can think of it as a fraction with a denominator of 1. For example, the reciprocal of \-8\ is \(\frac{1}{-8}\).

The reciprocal is necessary when converting a division problem into a multiplication problem. In the given problem \(-\frac{4}{3} \div\ \left(-\frac{1}{8}\right)\), you need to find the reciprocal of \left(-\frac{1}{8}\right)\, which is \-8\. Once you have the reciprocal, you can rewrite the division problem as a multiplication problem: \- \frac{4}{3} \times\ \-8\. This step simplifies the division of fractions.

Simplifying fractions ensures that your final answer is in its simplest form. To simplify a fraction means to reduce it to its smallest possible numerator and denominator. This can be done by dividing both the numerator and the denominator by their greatest common divisor (GCD).

However, sometimes a fraction is already in its simplest form, as in the given problem. When you reach \(\frac{32}{3}\), check if 32 and 3 have any common factors other than 1. They don't, so \(\frac{32}{3}\) is already simplified.

Remember, always try to simplify your fractions at the end of your calculation to ensure your answer is as clear and concise as possible.