benjamin writes an expression for the sum of 1 cubed, 2 cubed, and 3 cubed. What is the value of the expression
The correct answer and explanation is:
Correct Answer:
The expression for the sum of 1 cubed, 2 cubed, and 3 cubed is: 13+23+33=1+8+27=361^3 + 2^3 + 3^3 = 1 + 8 + 27 = \boxed{36}
Explanation (300 words):
To solve this problem, we need to evaluate the sum of three cube numbers. A cube number is the result of multiplying a number by itself three times. Mathematically, we express this as n3=n×n×nn^3 = n \times n \times n.
Benjamin writes the expression: 13+23+331^3 + 2^3 + 3^3
Let’s solve each term one at a time:
- 13=1×1×1=11^3 = 1 \times 1 \times 1 = 1
- 23=2×2×2=82^3 = 2 \times 2 \times 2 = 8
- 33=3×3×3=273^3 = 3 \times 3 \times 3 = 27
Now we add the results of each cube: 1+8+27=361 + 8 + 27 = 36
So, the value of the expression is 36.
This type of problem helps students understand how exponents work, particularly cubed numbers (also called “third powers”). It also demonstrates how math operations can be broken into steps—first by evaluating each cube and then adding them together.
An interesting fact is that the sum of the cubes of the first n natural numbers has a neat formula: 13+23+33+⋯+n3=(n(n+1)2)21^3 + 2^3 + 3^3 + \dots + n^3 = \left( \frac{n(n+1)}{2} \right)^2
If we apply it with n=3n = 3: (3(3+1)2)2=(122)2=62=36\left( \frac{3(3+1)}{2} \right)^2 = \left( \frac{12}{2} \right)^2 = 6^2 = 36
This confirms our answer. This formula is useful when working with larger sets of numbers.
In summary, Benjamin’s expression equals 36, and it teaches valuable concepts like exponentiation, order of operations, and number patterns in mathematics.