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Sagot :
Certainly! Let's evaluate the given expression step-by-step. The given mathematical expression is:
[tex]\[ \frac{-1}{6} \times \frac{4}{7} + \frac{1}{2} \times \frac{-3}{7} \times \frac{1}{6} \][/tex]
First, let's handle each fraction multiplication separately.
### First Term:
The first term is:
[tex]\[ \frac{-1}{6} \times \frac{4}{7} \][/tex]
Multiplying the numerators:
[tex]\[ -1 \times 4 = -4 \][/tex]
Multiplying the denominators:
[tex]\[ 6 \times 7 = 42 \][/tex]
So, the first term becomes:
[tex]\[ \frac{-4}{42} = \frac{-2}{21} \approx -0.09523809523809523 \][/tex]
### Second Term:
The second term involves calculating the product of three fractions. Let's first find:
[tex]\[ \frac{-3}{7} \times \frac{1}{6} \][/tex]
Multiplying the numerators:
[tex]\[ -3 \times 1 = -3 \][/tex]
Multiplying the denominators:
[tex]\[ 7 \times 6 = 42 \][/tex]
So, this part becomes:
[tex]\[ \frac{-3}{42} = \frac{-1}{14} \approx -0.07142857142857142 \][/tex]
Now we need to multiply this result by [tex]\(\frac{1}{2}\)[/tex]:
[tex]\[ \frac{1}{2} \times \frac{-1}{14} \][/tex]
Multiplying the numerators:
[tex]\[ 1 \times -1 = -1 \][/tex]
Multiplying the denominators:
[tex]\[ 2 \times 14 = 28 \][/tex]
So, this entire term becomes:
[tex]\[ \frac{-1}{28} \approx -0.03571428571428571 \][/tex]
### Adding the Two Terms:
Now we need to add the results of the two terms:
[tex]\[ \frac{-2}{21} + \frac{-1}{28} \][/tex]
First, let's convert them to a common denominator. The least common multiple of 21 and 28 is 84.
Converting [tex]\(\frac{-2}{21}\)[/tex]:
[tex]\[ \frac{-2}{21} = \frac{-2 \times 4}{21 \times 4} = \frac{-8}{84} \][/tex]
Converting [tex]\(\frac{-1}{28}\)[/tex]:
[tex]\[ \frac{-1}{28} = \frac{-1 \times 3}{28 \times 3} = \frac{-3}{84} \][/tex]
Now, adding these fractions:
[tex]\[ \frac{-8}{84} + \frac{-3}{84} = \frac{-8 - 3}{84} = \frac{-11}{84} \approx -0.13095238095238093 \][/tex]
Thus, the final result of the given expression is:
[tex]\[ \boxed{-0.13095238095238093} \][/tex]
This result matches our initial calculations.
[tex]\[ \frac{-1}{6} \times \frac{4}{7} + \frac{1}{2} \times \frac{-3}{7} \times \frac{1}{6} \][/tex]
First, let's handle each fraction multiplication separately.
### First Term:
The first term is:
[tex]\[ \frac{-1}{6} \times \frac{4}{7} \][/tex]
Multiplying the numerators:
[tex]\[ -1 \times 4 = -4 \][/tex]
Multiplying the denominators:
[tex]\[ 6 \times 7 = 42 \][/tex]
So, the first term becomes:
[tex]\[ \frac{-4}{42} = \frac{-2}{21} \approx -0.09523809523809523 \][/tex]
### Second Term:
The second term involves calculating the product of three fractions. Let's first find:
[tex]\[ \frac{-3}{7} \times \frac{1}{6} \][/tex]
Multiplying the numerators:
[tex]\[ -3 \times 1 = -3 \][/tex]
Multiplying the denominators:
[tex]\[ 7 \times 6 = 42 \][/tex]
So, this part becomes:
[tex]\[ \frac{-3}{42} = \frac{-1}{14} \approx -0.07142857142857142 \][/tex]
Now we need to multiply this result by [tex]\(\frac{1}{2}\)[/tex]:
[tex]\[ \frac{1}{2} \times \frac{-1}{14} \][/tex]
Multiplying the numerators:
[tex]\[ 1 \times -1 = -1 \][/tex]
Multiplying the denominators:
[tex]\[ 2 \times 14 = 28 \][/tex]
So, this entire term becomes:
[tex]\[ \frac{-1}{28} \approx -0.03571428571428571 \][/tex]
### Adding the Two Terms:
Now we need to add the results of the two terms:
[tex]\[ \frac{-2}{21} + \frac{-1}{28} \][/tex]
First, let's convert them to a common denominator. The least common multiple of 21 and 28 is 84.
Converting [tex]\(\frac{-2}{21}\)[/tex]:
[tex]\[ \frac{-2}{21} = \frac{-2 \times 4}{21 \times 4} = \frac{-8}{84} \][/tex]
Converting [tex]\(\frac{-1}{28}\)[/tex]:
[tex]\[ \frac{-1}{28} = \frac{-1 \times 3}{28 \times 3} = \frac{-3}{84} \][/tex]
Now, adding these fractions:
[tex]\[ \frac{-8}{84} + \frac{-3}{84} = \frac{-8 - 3}{84} = \frac{-11}{84} \approx -0.13095238095238093 \][/tex]
Thus, the final result of the given expression is:
[tex]\[ \boxed{-0.13095238095238093} \][/tex]
This result matches our initial calculations.
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