Question

multiply.
8·\frac{2}{5}·\frac{-9}{-3}
write your answer in simplest form.

Explanation:

Step1: Simplify the fractions

First, handle the negative signs and simplify \(\frac{-9}{-3}\). Since a negative divided by a negative is positive, and \(9\div3 = 3\), so \(\frac{-9}{-3}=3\).
Now the expression becomes \(8\cdot\frac{2}{5}\cdot3\).

Step2: Multiply the numbers

Multiply the whole number and the other numbers. First, multiply \(8\), \(2\), and \(3\) in the numerator: \(8\times2\times3 = 48\). The denominator remains \(5\) (since we had \(\frac{2}{5}\) and the other fraction simplified to a whole number, so we can think of it as \(\frac{8\times2\times3}{5}\)). Wait, actually, let's do it step by step. First, multiply \(8\) and \(\frac{2}{5}\): \(8\times\frac{2}{5}=\frac{16}{5}\). Then multiply \(\frac{16}{5}\) by \(3\) (which is \(\frac{3}{1}\)): \(\frac{16}{5}\times\frac{3}{1}=\frac{48}{5}\)? Wait, no, wait. Wait, the original expression is \(8\cdot\frac{2}{5}\cdot\frac{-9}{-3}\). Let's re - do it. The \(\frac{-9}{-3}\) is \(3\), so now we have \(8\times\frac{2}{5}\times3\). Let's multiply the integers first: \(8\times3 = 24\), then multiply by \(\frac{2}{5}\): \(24\times\frac{2}{5}=\frac{48}{5}\)? Wait, no, that's not right. Wait, no, the order of multiplication: \(8\times\frac{2}{5}\times\frac{-9}{-3}\). Let's handle the fractions first. \(\frac{-9}{-3}=3\), so now it's \(8\times\frac{2}{5}\times3\). Multiply \(8\) and \(3\) first: \(8\times3 = 24\), then \(24\times\frac{2}{5}=\frac{48}{5}\)? Wait, no, wait, maybe I made a mistake. Wait, let's do it with the fractions: \(8\) can be written as \(\frac{8}{1}\). So the expression is \(\frac{8}{1}\times\frac{2}{5}\times\frac{-9}{-3}\). First, simplify \(\frac{-9}{-3}=3=\frac{3}{1}\). Now we have \(\frac{8}{1}\times\frac{2}{5}\times\frac{3}{1}\). Multiply the numerators: \(8\times2\times3 = 48\). Multiply the denominators: \(1\times5\times1 = 5\). Wait, but that gives \(\frac{48}{5}\), but that's an improper fraction. Wait, no, wait, maybe I messed up the sign. Wait, \(\frac{-9}{-3}\) is positive \(3\), correct. Then \(8\times\frac{2}{5}\times3\). Let's calculate \(8\times3 = 24\), then \(24\times\frac{2}{5}=\frac{48}{5}=9.6\), but that seems odd. Wait, no, wait, maybe I made a mistake in the simplification. Wait, let's do it again. The original problem: \(8\cdot\frac{2}{5}\cdot\frac{-9}{-3}\). Let's simplify \(\frac{-9}{-3}\) first. \(\frac{-9}{-3}=3\), because \((-9)\div(-3)=3\). Now we have \(8\times\frac{2}{5}\times3\). Now, multiply \(8\) and \(3\) to get \(24\), then multiply \(24\) by \(\frac{2}{5}\): \(24\times\frac{2}{5}=\frac{48}{5}\). Wait, but \(\frac{48}{5}\) is \(9\frac{3}{5}\), but maybe I made a mistake. Wait, no, let's check the multiplication again. Wait, \(8\times\frac{2}{5}=\frac{16}{5}\), then \(\frac{16}{5}\times\frac{-9}{-3}\). \(\frac{-9}{-3}=3\), so \(\frac{16}{5}\times3=\frac{48}{5}\). Yes, that's correct. So the answer is \(\frac{48}{5}\) or \(9\frac{3}{5}\), but as an improper fraction, it's \(\frac{48}{5}\). Wait, but let's check the arithmetic again. \(8\times2 = 16\), \(16\times(-9)= - 144\), \(1\times5\times(-3)=-15\). Then \(\frac{-144}{-15}=\frac{144}{15}\). Simplify \(\frac{144}{15}\): divide numerator and denominator by \(3\), \(144\div3 = 48\), \(15\div3 = 5\). So \(\frac{48}{5}\). Ah, that's a better way. Multiply all numerators: \(8\times2\times(-9)=-144\), multiply all denominators: \(1\times5\times(-3)=-15\). Then \(\frac{-144}{-15}=\frac{144}{15}=\frac{48}{5}\) (dividing numerator and denominator by \(3\)). Yes, that's correct.

Answer:

\(\frac{48}{5}\)