Negative Substitution - Starting Points Maths - Free Printable

We are given:
a = -3, b = -1

We will evaluate each expression one by one, carefully substituting the values and following order of operations (PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction).

---

① a²

= (-3)² = 9

② 2b²

= 2 × (-1)² = 2 × 1 = 2

③ b³

= (-1)³ = -1

④ (-b)³

-b = -(-1) = 1 → (1)³ = 1

⑤ a + b

= -3 + (-1) = -4

⑥ a - b

= -3 - (-1) = -3 + 1 = -2

⑦ -a - b

-a = -(-3) = 3 → 3 - (-1) = 3 + 1 = 4? Wait — let’s read carefully:
It’s -a - b, which is (-a) minus b.
So: -(-3) - (-1) = 3 + 1 = 4 ✔

Wait — actually, standard interpretation:
“-a - b” means subtract a and then subtract b? No — it’s negative of a, minus b.

Yes: -a = 3, then 3 - b = 3 - (-1) = 4. Correct.

But let me double-check with direct substitution:

Expression: -a - b
Substitute: -(-3) - (-1) = 3 + 1 = 4 ✔

⑧ ab

= (-3)(-1) = 3

⑨ a² + b²

= 9 + 1 = 10

Ⓐ a² - b²

= 9 - 1 = 8

Ⓑ a³ + b³

a³ = (-3)³ = -27
b³ = (-1)³ = -1
→ -27 + (-1) = -28

Ⓒ a³ - b³

= -27 - (-1) = -27 + 1 = -26

Ⓓ 2a² + b

2×(9) + (-1) = 18 - 1 = 17

Ⓔ 2a² - b

18 - (-1) = 18 + 1 = 19

Ⓕ 2a² - 2b

= 2×9 - 2×(-1) = 18 + 2 = 20

Ⓖ 2a² - 2b²

= 18 - 2×1 = 18 - 2 = 16

Ⓗ 2a² - 2b³

b³ = -1 → 2b³ = -2
So: 18 - (-2) = 18 + 2 = 20

Wait — expression is 2a² - 2b³
= 2*(9) - 2*(-1) = 18 - (-2) = 20 ✔

Ⓘ 2b³ - 2a³

b³ = -1 → 2b³ = -2
a³ = -27 → 2a³ = -54
So: -2 - (-54) = -2 + 54 = 52

Ⓙ 2(a + b)

a + b = -4 → 2×(-4) = -8

Ⓚ 2(a - b)

a - b = -2 → 2×(-2) = -4

Ⓛ 3(b - a)

b - a = -1 - (-3) = -1 + 3 = 2 → 3×2 = 6

Ⓜ a(a - b)

a = -3, a - b = -2 → (-3)×(-2) = 6

Ⓝ b(b - a)

b = -1, b - a = 2 → (-1)×2 = -2

Ⓞ b(a - 2b)

First compute inside: a - 2b = -3 - 2*(-1) = -3 + 2 = -1
Then b × that = (-1) × (-1) = 1

Ⓟ a²(a + b²)

a² = 9, b² = 1 → a + b² = -3 + 1 = -2
Then 9 × (-2) = -18

Ⓠ b²(a² - b)

b² = 1, a² = 9, so a² - b = 9 - (-1) = 10
Then 1 × 10 = 10

Ⓡ (a + b)²

a + b = -4 → (-4)² = 16

Ⓢ (a - b)²

a - b = -2 → (-2)² = 4

Ⓣ (2a - b)²

2a = 2*(-3) = -6
-6 - b = -6 - (-1) = -5
(-5)² = 25

Ⓤ (a - b)³

a - b = -2 → (-2)³ = -8

Ⓥ a(a + b)²

a = -3, (a + b)² = 16 → -3 × 16 = -48

Ⓦ (2a² - 2b)²

From earlier: 2a² - 2b = 20 → 20² = 400

Wait — check:
2a² = 18, 2b = 2*(-1) = -2 → 2a² - 2b = 18 - (-2)? No!

Hold on! Expression is (2a² - 2b)²

2a² = 2*9 = 18
2b = 2*(-1) = -2
So 2a² - 2b = 18 - (-2)? NO — subtraction: 18 minus (2b)

2b is -2, so 18 - (-2) = 20? That’s correct for value, but let's write clearly:

Actually:
2a² - 2b = 2*(a²) - 2*(b) = 2*9 - 2*(-1) = 18 + 2 = 20 → yes.

Then squared: 20² = 400 ✔

Ⓧ a/b

= (-3)/(-1) = 3

Ⓨ (a + b)/b

a + b = -4, b = -1 → (-4)/(-1) = 4

Ⓩ (a - b)/b

a - b = -2, b = -1 → (-2)/(-1) = 2

α (a + b)/(a - b)

a + b = -4, a - b = -2 → (-4)/(-2) = 2

β (a² + b)/(a - b)

a² = 9, b = -1 → numerator: 9 + (-1) = 8
Denominator: a - b = -2
→ 8 / (-2) = -4

γ (a² + b)/(a - b)²

Numerator same: 8
Denominator: (a - b)² = (-2)² = 4
→ 8 / 4 = 2

Δ [2(a + b)] / [2(a + b)²]

Notice: we can simplify before plugging in.

Factor out 2:
= [2(a+b)] / [2(a+b)²] = 1 / (a+b) [as long as a+b ≠ 0]

a + b = -4 → 1 / (-4) = -0.25 or -1/4

Alternatively, plug in:

Numerator: 2*(-4) = -8
Denominator: 2*(-4)² = 2*16 = 32
→ -8 / 32 = -1/4 ✔

ε [2(b - a)] / [2(a - b)²]

Simplify first: factor 2 cancels → (b - a) / (a - b)²

Note: b - a = -(a - b)

So: [-(a - b)] / (a - b)² = -1 / (a - b)

a - b = -2 → -1 / (-2) = 1/2

Plug in to verify:

Numerator: 2*(b - a) = 2*[ -1 - (-3) ] = 2*(2) = 4
Denominator: 2*(a - b)² = 2*(-2)² = 2*4 = 8
→ 4 / 8 = 1/2 ✔

---

Now, compiling all answers in order:

① 9
② 2
③ -1
④ 1
⑤ -4
⑥ -2
⑦ 4
⑧ 3
⑨ 10
Ⓐ 8
Ⓑ -28
Ⓒ -26
Ⓓ 17
Ⓔ 19
Ⓕ 20
Ⓖ 16
Ⓗ 20
Ⓘ 52
Ⓙ -8
Ⓚ -4
Ⓛ 6
Ⓜ 6
Ⓝ -2
Ⓞ 1
Ⓟ -18
Ⓠ 10
Ⓡ 16
Ⓢ 4
Ⓣ 25
Ⓤ -8
Ⓥ -48
Ⓦ 400
Ⓧ 3
Ⓨ 4
Ⓩ 2
α 2
β -4
γ 2
Δ -1/4
ε 1/2

Let me double-check a few tricky ones:

- ⑦: -a - b = -(-3) - (-1) = 3 + 1 = 4 ✔
- Ⓗ: 2a² - 2b³ = 18 - 2*(-1) = 18 + 2 = 20 ✔
- Ⓘ: 2b³ - 2a³ = 2*(-1) - 2*(-27) = -2 + 54 = 52 ✔
- Ⓦ: (2a² - 2b)² = (18 - (-2))²? Wait no — 2a² - 2b = 18 - 2*(-1) = 18 + 2 = 20 → 400 ✔
- Δ: simplified to 1/(a+b) = 1/-4 = -1/4 ✔
- ε: simplified to -1/(a-b) = -1/-2 = 1/2 ✔

All look good.

Final Answer:
① 9
② 2
③ -1
④ 1
⑤ -4
⑥ -2
⑦ 4
⑧ 3
⑨ 10
Ⓐ 8
Ⓑ -28
Ⓒ -26
Ⓓ 17
Ⓔ 19
Ⓕ 20
Ⓖ 16
Ⓗ 20
Ⓘ 52
Ⓙ -8
Ⓚ -4
Ⓛ 6
Ⓜ 6
Ⓝ -2
Ⓞ 1
Ⓟ -18
Ⓠ 10
Ⓡ 16
Ⓢ 4
Ⓣ 25
Ⓤ -8
Ⓥ -48
Ⓦ 400
Ⓧ 3
Ⓨ 4
Ⓩ 2
α 2
β -4
γ 2
Δ -1/4
ε 1/2