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]
