$\left(-\frac{3}{4}\right)\left(\frac{7}{8}\right)$
The expression given is the product of two fractions: \(-\frac{3}{4}\) and \(\frac{7}{8}\). Our goal is to perform this multiplication according to the standard rules for rational numbers. We note that the first fraction is negative and the second fraction is positive, so the final result will be negative. The rule for multiplying two fractions \(\frac{a}{b}\) and \(\frac{c}{d}\) is given by \(\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}\).
To multiply the fractions, we multiply the numerators together and the denominators together separately:
Combining these, we get:
\[\left(-\frac{3}{4}\right) \cdot \left(\frac{7}{8}\right) = -\frac{3 \times 7}{4 \times 8} = -\frac{21}{32}\]
We now check if the fraction \(-\frac{21}{32}\) can be simplified further. The prime factors of 21 are 3 and 7, while the prime factors of 32 are all 2s (\(2^5\)). Since there are no common factors between the numerator and the denominator, the fraction is already in its simplest form. Thus, the final product is \(-\frac{21}{32}\).
Multiplying fractions is useful in many everyday situations, such as cooking. Imagine a recipe calls for \(\frac{3}{4}\) of a specific seasoning, but you decide to scale the recipe down to \(\frac{7}{8}\) of the original size because you are only cooking for a few people. To find the amount of seasoning you need, you would calculate \(\frac{3}{4} \times \frac{7}{8} = \frac{21}{32}\) of the unit of measurement. Understanding how to multiply fractions ensures you can accurately adjust quantities in recipes, construction plans, or financial distributions.