Algebra worksheet with simplifying expressions and a pyramid problem.

Let's go through each section of the worksheet and solve the problems step by step.

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Starter Questions

Simplify the expressions below:

1) $ a + a + a $
→ Combine like terms: $ 3a $

2) $ 2p + 3p + 5p $
→ $ (2+3+5)p = 10p $

3) $ 7u - 3u $
→ $ (7-3)u = 4u $

4) $ 8r + 11r - 4r $
→ $ (8+11-4)r = 15r $

✔ Starter Answers:
1) $ 3a $
2) $ 10p $
3) $ 4u $
4) $ 15r $

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Level 1

Simplify the expressions below:

1) $ 3a + 4a + 3b + 2b $
→ $ (3a+4a) + (3b+2b) = 7a + 5b $

2) $ 5t + 3t + 5v + 11v $
→ $ (5t+3t) + (5v+11v) = 8t + 16v $

3) $ 7u - 3u + 6k - 2k $
→ $ (7u-3u) + (6k-2k) = 4u + 4k $

4) $ 5f - f + 65w - 23w $
→ $ (5f-f) + (65w-23w) = 4f + 42w $

5) $ 6h - 4h + 4d - 8d $
→ $ (6h-4h) + (4d-8d) = 2h - 4d $

6) $ 7a + 4 + 3a + 2 $
→ $ (7a+3a) + (4+2) = 10a + 6 $

7) $ 8t + 3 + 6t + v $
→ $ (8t+6t) + 3 + v = 14t + v + 3 $

8) $ 7q + 3u - 6q - 2u $
→ $ (7q-6q) + (3u-2u) = q + u $

9) $ 12f - 8 - 8f - 7 $
→ $ (12f-8f) + (-8-7) = 4f - 15 $

10) $ 6h - 5d - 9h + 5d $
→ $ (6h-9h) + (-5d+5d) = -3h + 0 = -3h $

✔ Level 1 Answers:
1) $ 7a + 5b $
2) $ 8t + 16v $
3) $ 4u + 4k $
4) $ 4f + 42w $
5) $ 2h - 4d $
6) $ 10a + 6 $
7) $ 14t + v + 3 $
8) $ q + u $
9) $ 4f - 15 $
10) $ -3h $

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Legend Questions

Simplify the expressions below:

1) $ 5aa + a^2 - a + 6ab - b + bb $
Note: $ aa = a^2 $, $ bb = b^2 $
So:
$ 5a^2 + a^2 - a + 6ab - b + b^2 $
→ Combine like terms:
$ (5a^2 + a^2) + 6ab + b^2 - a - b $
= $ 6a^2 + 6ab + b^2 - a - b $

2) $ 6tr - tr^2 + (tr)^2 - 3t^2r^2 - ttrr $
Note:
- $ tr^2 = t \cdot r^2 $
- $ (tr)^2 = t^2r^2 $
- $ ttrr = t^2r^2 $

So:
$ 6tr - tr^2 + t^2r^2 - 3t^2r^2 - t^2r^2 $
Now combine:
- $ 6tr $ → only one term
- $ -tr^2 $ → only one
- $ t^2r^2 - 3t^2r^2 - t^2r^2 = (1 - 3 - 1)t^2r^2 = -3t^2r^2 $

Final: $ 6tr - tr^2 - 3t^2r^2 $

3) $ a(-a)^2 - cb^2 + 6c^2b + 5 - 7a^3 $
First: $ (-a)^2 = a^2 $, so $ a \cdot a^2 = a^3 $
So:
$ a^3 - cb^2 + 6c^2b + 5 - 7a^3 $
Combine like terms:
$ (a^3 - 7a^3) + (-cb^2) + 6c^2b + 5 $
= $ -6a^3 - cb^2 + 6c^2b + 5 $

4) $ 5x^4(3x^2 - 2)^2 - (x - 2)^3 $
This is more complex. Let’s expand both parts.

First: $ (3x^2 - 2)^2 = (3x^2)^2 - 2\cdot3x^2\cdot2 + 2^2 = 9x^4 - 12x^2 + 4 $

Now multiply by $ 5x^4 $:
$ 5x^4(9x^4 - 12x^2 + 4) = 45x^8 - 60x^6 + 20x^4 $

Second: $ (x - 2)^3 = x^3 - 3x^2(2) + 3x(4) - 8 = x^3 - 6x^2 + 12x - 8 $

Now subtract:
$ (45x^8 - 60x^6 + 20x^4) - (x^3 - 6x^2 + 12x - 8) $
= $ 45x^8 - 60x^6 + 20x^4 - x^3 + 6x^2 - 12x + 8 $

✔ Legend Answers:
1) $ 6a^2 + 6ab + b^2 - a - b $
2) $ 6tr - tr^2 - 3t^2r^2 $
3) $ -6a^3 - cb^2 + 6c^2b + 5 $
4) $ 45x^8 - 60x^6 + 20x^4 - x^3 + 6x^2 - 12x + 8 $

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Problem Solvers – Brick Puzzle

Each brick is the sum of the two bricks below it.

We have this pyramid:

```
[ ? ]
/ \
[6x - 19] [ ? ]
/ \ / \
[5x² + 4] [x² + x] [5x² - 12x + 57]
```

We need to fill in the missing boxes.

#### Step 1: Find the right middle brick

It's the sum of the bottom two on the right:
$ (x^2 + x) + (5x^2 - 12x + 57) $
= $ x^2 + 5x^2 + x - 12x + 57 $
= $ 6x^2 - 11x + 57 $

So the right middle brick is $ 6x^2 - 11x + 57 $

#### Step 2: Find the top brick

It's the sum of the two middle bricks:
$ (6x - 19) + (6x^2 - 11x + 57) $
= $ 6x^2 + (6x - 11x) + (-19 + 57) $
= $ 6x^2 - 5x + 38 $

✔ So the top brick is $ 6x^2 - 5x + 38 $

Now fill in the blanks:

- Middle right brick: $ 6x^2 - 11x + 57 $
- Top brick: $ 6x^2 - 5x + 38 $

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✔ Final Answers Summary

#### Starter Questions:
1) $ 3a $
2) $ 10p $
3) $ 4u $
4) $ 15r $

#### Level 1:
1) $ 7a + 5b $
2) $ 8t + 16v $
3) $ 4u + 4k $
4) $ 4f + 42w $
5) $ 2h - 4d $
6) $ 10a + 6 $
7) $ 14t + v + 3 $
8) $ q + u $
9) $ 4f - 15 $
10) $ -3h $

#### Legend Questions:
1) $ 6a^2 + 6ab + b^2 - a - b $
2) $ 6tr - tr^2 - 3t^2r^2 $
3) $ -6a^3 - cb^2 + 6c^2b + 5 $
4) $ 45x^8 - 60x^6 + 20x^4 - x^3 + 6x^2 - 12x + 8 $

#### Problem Solvers:
- Middle right brick: $ 6x^2 - 11x + 57 $
- Top brick: $ 6x^2 - 5x + 38 $

Let me know if you'd like these formatted as a printable answer sheet!