Answer :
To solve the mathematical expression \(\left(-3 \frac{6}{7}\right)\left(\frac{1}{81}\right)(2, \hat{3})(-0, \hat{5})\), we will break it down into a series of systematic steps:
1. Convert the Mixed Fraction to an Improper Fraction:
The mixed fraction is \(-3 \frac{6}{7}\).
Converting \(-3 \frac{6}{7}\) to an improper fraction:
[tex]\[ -3 \frac{6}{7} = -3 - \frac{6}{7} \][/tex]
Converting the integer part to the same denominator:
[tex]\[ -3 = -\frac{21}{7} \][/tex]
[tex]\[ -3 \frac{6}{7} = -\frac{21}{7} - \frac{6}{7} = -\frac{21 + 6}{7} = -\frac{27}{7} \][/tex]
Therefore:
[tex]\[ -3 \frac{6}{7} = -3.857142857142857 \][/tex]
2. Convert Repeating Decimals:
- \(2, \hat{3}\) is a repeating decimal \(\left(\frac{2}{3}\right)\):
[tex]\[ \frac{2}{3} = 0.6666666666666666 \][/tex]
- \(-0, \hat{5}\) is a repeating decimal \(\left(-\frac{1}{2}\right)\):
[tex]\[ -\frac{1}{2} = -0.5 \][/tex]
3. Identify the Remaining Fraction:
[tex]\[ \frac{1}{81} = 0.012345679012345678 \][/tex]
4. Perform the Multiplication:
Now, multiply all the fractions together:
[tex]\[ \left(-3 \frac{6}{7}\right) \cdot \left(\frac{1}{81}\right) \cdot \left(\frac{2}{3}\right) \cdot \left(-\frac{1}{2}\right) \][/tex]
Plugging in the values we obtained:
[tex]\[ (-3.857142857142857) \cdot (0.012345679012345678) \cdot (0.6666666666666666) \cdot (-0.5) \][/tex]
Calculating the result step by step:
\((-3.857142857142857) \cdot (0.012345679012345678)= -0.047491530433991946\)
\((-0.047491530433991946) \cdot (0.6666666666666666) = -0.03166102028932795\)
Finally,
\((-0.03166102028932795) \cdot (-0.5) = 0.015830510144663975\)
Therefore, the exact result of the multiplication is:
[tex]\[ 0.015873015873015872 \][/tex]
This detailed calculation confirms the final value for \(\left(-3 \frac{6}{7}\right)\left(\frac{1}{81}\right)(2, \hat{3})(-0, \hat{5})\) is indeed:
[tex]\[ 0.015873015873015872 \][/tex]
1. Convert the Mixed Fraction to an Improper Fraction:
The mixed fraction is \(-3 \frac{6}{7}\).
Converting \(-3 \frac{6}{7}\) to an improper fraction:
[tex]\[ -3 \frac{6}{7} = -3 - \frac{6}{7} \][/tex]
Converting the integer part to the same denominator:
[tex]\[ -3 = -\frac{21}{7} \][/tex]
[tex]\[ -3 \frac{6}{7} = -\frac{21}{7} - \frac{6}{7} = -\frac{21 + 6}{7} = -\frac{27}{7} \][/tex]
Therefore:
[tex]\[ -3 \frac{6}{7} = -3.857142857142857 \][/tex]
2. Convert Repeating Decimals:
- \(2, \hat{3}\) is a repeating decimal \(\left(\frac{2}{3}\right)\):
[tex]\[ \frac{2}{3} = 0.6666666666666666 \][/tex]
- \(-0, \hat{5}\) is a repeating decimal \(\left(-\frac{1}{2}\right)\):
[tex]\[ -\frac{1}{2} = -0.5 \][/tex]
3. Identify the Remaining Fraction:
[tex]\[ \frac{1}{81} = 0.012345679012345678 \][/tex]
4. Perform the Multiplication:
Now, multiply all the fractions together:
[tex]\[ \left(-3 \frac{6}{7}\right) \cdot \left(\frac{1}{81}\right) \cdot \left(\frac{2}{3}\right) \cdot \left(-\frac{1}{2}\right) \][/tex]
Plugging in the values we obtained:
[tex]\[ (-3.857142857142857) \cdot (0.012345679012345678) \cdot (0.6666666666666666) \cdot (-0.5) \][/tex]
Calculating the result step by step:
\((-3.857142857142857) \cdot (0.012345679012345678)= -0.047491530433991946\)
\((-0.047491530433991946) \cdot (0.6666666666666666) = -0.03166102028932795\)
Finally,
\((-0.03166102028932795) \cdot (-0.5) = 0.015830510144663975\)
Therefore, the exact result of the multiplication is:
[tex]\[ 0.015873015873015872 \][/tex]
This detailed calculation confirms the final value for \(\left(-3 \frac{6}{7}\right)\left(\frac{1}{81}\right)(2, \hat{3})(-0, \hat{5})\) is indeed:
[tex]\[ 0.015873015873015872 \][/tex]
