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 expression (45)3\left( \frac{4}{5} \right)^3(54)3 represents the cube of the fraction 45\frac{4}{5}54. Let’s break down the components of this operation and clarify the confusion with the options.
Step 1: Understanding the operation
When we cube a fraction, we multiply the fraction by itself three times. This gives:(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 2: Multiplying the fractions
To multiply fractions, multiply the numerators and denominators separately: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
Now, multiply this result by the third 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, the correct value of (45)3\left( \frac{4}{5} \right)^3(54)3 is 64125\frac{64}{125}12564.
Step 3: Explanation of the incorrect options
- Option 1: 45+45+45=125\frac{4}{5} + \frac{4}{5} + \frac{4}{5} = \frac{12}{5}54+54+54=512
This is a sum, not a product. When you add 45\frac{4}{5}54 three times, you get 125\frac{12}{5}512, but this is not the correct operation for cubing the fraction. - Option 2: 45⋅45⋅45=64125\frac{4}{5} \cdot \frac{4}{5} \cdot \frac{4}{5} = \frac{64}{125}54⋅54⋅54=12564
This is the correct result and matches the correct calculation we did above. - Option 3: 4+4+45=125\frac{4 + 4 + 4}{5} = \frac{12}{5}54+4+4=512
This is again a sum in the numerator, not a product of fractions. It doesn’t correctly represent cubing the fraction. - Option 4: 54⋅54⋅54=12564\frac{5}{4} \cdot \frac{5}{4} \cdot \frac{5}{4} = \frac{125}{64}45⋅45⋅45=64125
This operation involves cubing 54\frac{5}{4}45, which is a different fraction than 45\frac{4}{5}54, so it’s irrelevant to the original problem.
Final Answer:
The correct expanded form and value of (45)3\left( \frac{4}{5} \right)^3(54)3 is:64125\frac{64}{125}12564
