March Math Worksheets 7th Grade - Free Printable

To solve this "Algebra Calculation Game," we need to simplify the algebraic expressions found in each hexagon. Since there is no specific value given for $n$ (like $n=2$), the goal is to rewrite each expression in its simplest form by expanding brackets and combining like terms.

Here is the step-by-step simplification for every expression on the board, organized by their position to help you check your work.

Top Row

1. $n^3$: This is already simplified.
* Result: $n^3$
2. $6n - 2$: This is already simplified.
* Result: $6n - 2$

Second Row from Top

3. $3n - 5$: Already simplified.
* Result: $3n - 5$
4. $2n - 5$: Already simplified.
* Result: $2n - 5$
5. $3(n - 1)$: Distribute the 3 into the parentheses.
* $3 \times n = 3n$
* $3 \times -1 = -3$
* Result: $3n - 3$

Third Row from Top

6. $10 - 3n$: Already simplified (usually written as $-3n + 10$, but this is fine).
* Result: $10 - 3n$
7. $\frac{1}{2}(n + 1)$: Distribute the $\frac{1}{2}$ (or divide by 2).
* $\frac{1}{2} \times n = \frac{n}{2}$ or $0.5n$
* $\frac{1}{2} \times 1 = \frac{1}{2}$ or $0.5$
* Result: $\frac{n}{2} + \frac{1}{2}$ (or $0.5n + 0.5$)
8. $n^2 - 3$: Already simplified.
* Result: $n^2 - 3$

Fourth Row from Top (Middle Area)

9. $2n^2$: Already simplified.
* Result: $2n^2$
10. $n^2 + 6$: Already simplified.
* Result: $n^2 + 6$
11. $3(n + 2)$: Distribute the 3.
* $3 \times n = 3n$
* $3 \times 2 = 6$
* Result: $3n + 6$
12. $2(n - 3)$: Distribute the 2.
* $2 \times n = 2n$
* $2 \times -3 = -6$
* Result: $2n - 6$

Fifth Row from Top

13. $10 - n^2$: Already simplified (usually written as $-n^2 + 10$).
* Result: $10 - n^2$
14. $n(n + 2)$: Distribute the $n$.
* $n \times n = n^2$
* $n \times 2 = 2n$
* Result: $n^2 + 2n$
15. $5 - 2n$: Already simplified.
* Result: $5 - 2n$

Bottom Row

16. $(n + 3)^2$: Expand the squared binomial $(n+3)(n+3)$.
* $n \times n = n^2$
* $n \times 3 = 3n$
* $3 \times n = 3n$
* $3 \times 3 = 9$
* Combine like terms ($3n + 3n = 6n$):
* Result: $n^2 + 6n + 9$
17. $6n - n^2$: Already simplified (usually written as $-n^2 + 6n$).
* Result: $6n - n^2$
18. $2(2n - 1)$: Distribute the 2.
* $2 \times 2n = 4n$
* $2 \times -1 = -2$
* Result: $4n - 2$
19. $n^2 - 2n$: Already simplified.
* Result: $n^2 - 2n$
20. $(n - 2)^2$: Expand the squared binomial $(n-2)(n-2)$.
* $n \times n = n^2$
* $n \times -2 = -2n$
* $-2 \times n = -2n$
* $-2 \times -2 = +4$
* Combine like terms ($-2n - 2n = -4n$):
* Result: $n^2 - 4n + 4$

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Summary of Simplified Expressions

If you need to match equivalent expressions or just list the final simplified forms, here they are:

* $n^3$
* $6n - 2$
* $3n - 5$
* $2n - 5$
* $3n - 3$ *(from $3(n-1)$)*
* $10 - 3n$
* $0.5n + 0.5$ *(from $\frac{1}{2}(n+1)$)*
* $n^2 - 3$
* $2n^2$
* $n^2 + 6$
* $3n + 6$ *(from $3(n+2)$)*
* $2n - 6$ *(from $2(n-3)$)*
* $10 - n^2$
* $n^2 + 2n$ *(from $n(n+2)$)*
* $5 - 2n$
* $n^2 + 6n + 9$ *(from $(n+3)^2$)*
* $6n - n^2$
* $4n - 2$ *(from $2(2n-1)$)*
* $n^2 - 2n$
* $n^2 - 4n + 4$ *(from $(n-2)^2$)*

Final Answer: The task requires simplifying each algebraic expression. The simplified forms are listed above in the summary. For example, $3(n-1)$ becomes $3n-3$, $(n+3)^2$ becomes $n^2+6n+9$, and $n(n+2)$ becomes $n^2+2n$.