Which is equivalent to $\left(9^3-6^3\right)$?
A. 45
B. 216
C. 513
D. 729
Answer (2)
Calculate 9 3 = 729 .
Calculate 6 3 = 216 .
Subtract the values: 729 − 216 = 513 .
The equivalent value is 513 .
Explanation
Understanding the Problem We are asked to find the value of the expression 9 3 − 6 3 and determine which of the given options is equivalent to it.
Calculating 9 cubed First, we need to calculate 9 3 . This means 9 × 9 × 9 . We have 9 × 9 = 81 , and then 81 × 9 = 729 . So, 9 3 = 729 .
Calculating 6 cubed Next, we need to calculate 6 3 . This means 6 × 6 × 6 . We have 6 × 6 = 36 , and then 36 × 6 = 216 . So, 6 3 = 216 .
Subtracting the values Now, we subtract 6 3 from 9 3 , which is 729 − 216 . Performing the subtraction, we get 729 − 216 = 513 .
Finding the equivalent option Therefore, 9 3 − 6 3 = 513 . Comparing this result with the given options, we find that option C, 513, matches our result.
Examples
Understanding the difference of cubes, like 9 3 − 6 3 , is useful in various fields. For example, in engineering, calculating the volume difference between two cubes of different sizes is a common task. If you have a cube-shaped container with a side length of 9 units and you remove a smaller cube with a side length of 6 units from it, the remaining volume is 9 3 − 6 3 = 513 cubic units. This concept also applies to architecture, where designers might calculate material quantities for structures with cubic elements.
The value of 9 3 − 6 3 is 513 . Therefore, the correct answer is option C. This was determined by calculating each cube and then performing the subtraction.
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