What is the correct expanded form and value of \left(\frac{4}{5}\right)^3? \ ? \frac{4}{5} + \frac{4}{5} + \frac{4}{5} = \frac{12}{5} \ ? \frac{4}{5} \cdot \frac{4}{5} \cdot \frac{4}{5} = \frac{64}{125} \ ? \frac{4 + 4 + 4}{5} = \frac{12}{5} \ ? \frac{5}{4} \cdot \frac{5}{4} \cdot \frac{5}{4} = \frac{125}{64}
The Correct Answer and Explanation is:
The correct expanded form and value of (45)3\left(\frac{4}{5}\right)^3(54)3 is given by the product of 45\frac{4}{5}54 repeated three times:(45)3=45⋅45⋅45\left(\frac{4}{5}\right)^3 = \frac{4}{5} \cdot \frac{4}{5} \cdot \frac{4}{5}(54)3=54⋅54⋅54
Step-by-Step Calculation:
- Multiplication of fractions: To multiply fractions, multiply the numerators together and the denominators together: 45⋅45=4⋅45⋅5=1625\frac{4}{5} \cdot \frac{4}{5} = \frac{4 \cdot 4}{5 \cdot 5} = \frac{16}{25}54⋅54=5⋅54⋅4=2516
- Multiplying again by 45\frac{4}{5}54: 1625⋅45=16⋅425⋅5=64125\frac{16}{25} \cdot \frac{4}{5} = \frac{16 \cdot 4}{25 \cdot 5} = \frac{64}{125}2516⋅54=25⋅516⋅4=12564
Thus, (45)3=64125\left(\frac{4}{5}\right)^3 = \frac{64}{125}(54)3=12564.
Explanation of Other Options:
- 45+45+45=125\frac{4}{5} + \frac{4}{5} + \frac{4}{5} = \frac{12}{5}54+54+54=512: This is an addition of three fractions, not an exponentiation. The result of adding 45\frac{4}{5}54 three times is 125\frac{12}{5}512, but this is unrelated to (45)3\left(\frac{4}{5}\right)^3(54)3.
- 4+4+45=125\frac{4 + 4 + 4}{5} = \frac{12}{5}54+4+4=512: This expression represents a sum in the numerator of the fraction and is also an addition problem. The correct result is 125\frac{12}{5}512, but again, this does not correspond to the cube of 45\frac{4}{5}54.
- 54⋅54⋅54=12564\frac{5}{4} \cdot \frac{5}{4} \cdot \frac{5}{4} = \frac{125}{64}45⋅45⋅45=64125: This is a multiplication of 54\frac{5}{4}45 three times, which results in 12564\frac{125}{64}64125, but it is not related to the expression (45)3\left(\frac{4}{5}\right)^3(54)3.
Conclusion:
The correct answer for the expanded form and value of (45)3\left(\frac{4}{5}\right)^3(54)3 is:64125\frac{64}{125}12564
