Question

fractions
signed fraction multiplication: advanced
multiply.
\\(\frac{3}{-4}\cdot(-7)\cdot\frac{-2}{9}\\)
write your answer in simplest form.

Explanation:

Step1: Simplify signs and multiply numerators/denominators

First, handle the signs: a negative times a negative is positive, and we have one negative from \(\frac{3}{-4}\) (which is \(-\frac{3}{4}\)) and a negative from \(-7\) and a negative from \(\frac{-2}{9}\) (wait, actually, let's rewrite the fractions with correct signs. \(\frac{3}{-4}=-\frac{3}{4}\), \(-7 = -\frac{7}{1}\), \(\frac{-2}{9}=-\frac{2}{9}\). Now, multiplying three numbers: two negatives (from \(-\frac{3}{4}\) and \(-\frac{2}{9}\)) and one negative (from \(-7\))? Wait, no: \(-\frac{3}{4} \cdot (-7) \cdot (-\frac{2}{9})\). Wait, the original is \(\frac{3}{-4} \cdot (-7) \cdot \frac{-2}{9}\). Let's rewrite all with signs: \(\frac{3}{-4}=-\frac{3}{4}\), \(-7\) is \(-7\), \(\frac{-2}{9}=-\frac{2}{9}\). So multiplying: \((-\frac{3}{4}) \cdot (-7) \cdot (-\frac{2}{9})\). The number of negative signs: three (since \(-\frac{3}{4}\) has one, \(-7\) has one, \(-\frac{2}{9}\) has one). Three negatives make a negative. Now multiply numerators: \(3 \cdot 7 \cdot 2 = 42\). Multiply denominators: \(4 \cdot 1 \cdot 9 = 36\). Wait, but let's do it step by step to simplify earlier.

Alternative step: First, multiply the fractions and the integer. Let's rewrite the expression as \(\frac{3}{-4} \times (-7) \times \frac{-2}{9}\). Let's cancel common factors first. The 3 in the numerator and 9 in the denominator: 3 and 9 have a common factor of 3. So 3 ÷ 3 = 1, 9 ÷ 3 = 3. Now the expression becomes \(\frac{1}{-4} \times (-7) \times \frac{-2}{3}\). Now, multiply the numerators: 1 × (-7) × (-2) = 14? Wait, no: wait, the signs: \(\frac{1}{-4}\) is \(-\frac{1}{4}\), times \(-7\) is positive \(\frac{7}{4}\), times \(-\frac{2}{3}\) is negative \(\frac{14}{12}\). Wait, maybe better to handle signs first. The number of negative factors: \(\frac{3}{-4}\) is negative (1 negative), \(-7\) is negative (2nd negative), \(\frac{-2}{9}\) is negative (3rd negative). So total negative signs: 3, which is odd, so the result is negative. Now multiply the absolute values: \(\frac{3}{4} \times 7 \times \frac{2}{9}\). Now, simplify \(\frac{3}{9}\) to \(\frac{1}{3}\). So now we have \(\frac{1}{4} \times 7 \times \frac{2}{3}\). Now, multiply \(\frac{1 \times 7 \times 2}{4 \times 3} = \frac{14}{12}\). Simplify \(\frac{14}{12}\) to \(\frac{7}{6}\). But since we had three negative signs, the result is negative? Wait, no: wait, original expression: \(\frac{3}{-4} \cdot (-7) \cdot \frac{-2}{9}\). Let's count the negative signs: \(\frac{3}{-4}\) has one negative (in denominator), \(-7\) has one negative, \(\frac{-2}{9}\) has one negative (in numerator). So total negative signs: 1 (from denominator) + 1 (from -7) + 1 (from numerator) = 3? Wait, no: a fraction \(\frac{a}{-b}\) is equal to \(-\frac{a}{b}\), and \(\frac{-a}{b}\) is also \(-\frac{a}{b}\). So \(\frac{3}{-4} = -\frac{3}{4}\), \(\frac{-2}{9} = -\frac{2}{9}\), and \(-7 = -7\). So the expression is \(-\frac{3}{4} \times -7 \times -\frac{2}{9}\). Now, multiplying two negatives: \(-\frac{3}{4} \times -7 = \frac{21}{4}\). Then multiply by \(-\frac{2}{9}\): \(\frac{21}{4} \times -\frac{2}{9} = -\frac{42}{36}\). Simplify \(-\frac{42}{36}\) by dividing numerator and denominator by 6: \(-\frac{7}{6}\). Wait, that's different from before. Wait, let's do it with canceling:

\(\frac{3}{-4} \cdot (-7) \cdot \frac{-2}{9}\)

First, rewrite all with signs:

\(= \frac{3}{-4} \times (-7) \times \frac{-2}{9}\)

\(= \frac{3 \times (-7) \times (-2)}{-4 \times 1 \times 9}\) (since -7 is \(\frac{-7}{1}\))

Now, numerator: 3 × (-7) × (-2) = 3 × 14 = 42 (because two negatives multiply to positive)

Denominator: -4 × 1 × 9 = -36

So the fraction is \(\frac{42}{-36} = -\frac{42}{36}\)

Simplify by dividing numerator and denominator by 6: \(-\frac{7}{6}\)

Yes, that's correct. So the steps:

Step1: Multiply numerators and denominators

Multiply all numerators: \(3 \times (-7) \times (-2) = 3 \times 14 = 42\) (since \((-7) \times (-2) = 14\))
Multiply all denominators: \(-4 \times 1 \times 9 = -36\) (since -7 is \(\frac{-7}{1}\), so denominator is 1)
So the fraction is \(\frac{42}{-36}\)

Step2: Simplify the fraction

Simplify \(\frac{42}{-36}\) by dividing numerator and denominator by their greatest common divisor (GCD), which is 6.
\(\frac{42 \div 6}{-36 \div 6} = \frac{7}{-6} = -\frac{7}{6}\)

Answer:

\(-\frac{7}{6}\)