Subject enrichment activity: Complete the diagram with suitable values so that each expression equals 5⁹.

We are given a central value of $ 5^9 $, and we need to complete each expression around it so that each one simplifies to $ 5^9 $. We'll use the laws of exponents to solve for the missing values (represented by boxes $ \square $) in each case.

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🔷 Laws of Exponents Used:

1. $ a^m \times a^n = a^{m+n} $
2. $ \frac{a^m}{a^n} = a^{m-n} $
3. $ (a^m)^n = a^{m \cdot n} $
4. $ a^0 = 1 $
5. $ a^{m/n} = \sqrt[n]{a^m} $
6. $ (ab)^n = a^n b^n $, but here only base 5
7. $ 25 = 5^2 $

Let’s go through each expression one by one.

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🟥 Red Circle: $ \dfrac{5^{18}}{5^\square} = 5^9 $

Using: $ \frac{5^{18}}{5^x} = 5^{18 - x} $

Set equal to $ 5^9 $:

$$
18 - x = 9 \Rightarrow x = 9
$$

✔ Answer: $ \boxed{9} $

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🟦 Cyan Circle: $ 5^\square \times 5^5 = 5^9 $

$ 5^x \times 5^5 = 5^{x+5} $

Set equal to $ 5^9 $:

$$
x + 5 = 9 \Rightarrow x = 4
$$

✔ Answer: $ \boxed{4} $

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🔵 Blue Circle: $ 5^{3(\square)} = 5^9 $

So: $ 5^{3x} = 5^9 $

$$
3x = 9 \Rightarrow x = 3
$$

✔ Answer: $ \boxed{3} $

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🟩 Green Circle: $ 5^{12} \div 5^\square = 5^9 $

$ \frac{5^{12}}{5^x} = 5^{12 - x} $

Set equal to $ 5^9 $:

$$
12 - x = 9 \Rightarrow x = 3
$$

✔ Answer: $ \boxed{3} $

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🟨 Orange Circle: $ 5^7 \times 5^\square = 5^9 $

$ 5^{7 + x} = 5^9 $

$$
7 + x = 9 \Rightarrow x = 2
$$

✔ Answer: $ \boxed{2} $

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🟪 Purple Circle: $ 5^0 \times 5^\square = 5^9 $

Note: $ 5^0 = 1 $, so this becomes:

$ 1 \times 5^x = 5^x $

Set equal to $ 5^9 $ → $ x = 9 $

✔ Answer: $ \boxed{9} $

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🟫 Brown Circle: $ 25^2 \times 5^\square = 5^9 $

First, $ 25 = 5^2 $, so $ 25^2 = (5^2)^2 = 5^{4} $

So expression becomes:

$ 5^4 \times 5^x = 5^{4+x} $

Set equal to $ 5^9 $:

$$
4 + x = 9 \Rightarrow x = 5
$$

✔ Answer: $ \boxed{5} $

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🟡 Yellow Circle: $ \dfrac{5^6 \times 5^\square}{5^2} = 5^9 $

Numerator: $ 5^6 \times 5^x = 5^{6+x} $

Divide by $ 5^2 $: $ 5^{6+x-2} = 5^{4+x} $

Set equal to $ 5^9 $:

$$
4 + x = 9 \Rightarrow x = 5
$$

✔ Answer: $ \boxed{5} $

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🟧 Reddish Orange Circle: $ (5^\square)^{3/2} = 5^9 $

Let $ x $ be the exponent: $ (5^x)^{3/2} = 5^{x \cdot \frac{3}{2}} $

Set equal to $ 5^9 $:

$$
\frac{3}{2}x = 9 \Rightarrow x = \frac{9 \times 2}{3} = 6
$$

✔ Answer: $ \boxed{6} $

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🟩 Light Green Circle: $ (5^\square)^9 = 5^9 $

Let $ x $ be the exponent: $ (5^x)^9 = 5^{9x} $

Set equal to $ 5^9 $:

$$
9x = 9 \Rightarrow x = 1
$$

✔ Answer: $ \boxed{1} $

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✔ Final Answers Summary:

| Circle | Expression | Missing Value |
|-------|------------|----------------|
| Red | $ \dfrac{5^{18}}{5^\square} $ | 9 |
| Cyan | $ 5^\square \times 5^5 $ | 4 |
| Blue | $ 5^{3(\square)} $ | 3 |
| Green | $ 5^{12} \div 5^\square $ | 3 |
| Orange | $ 5^7 \times 5^\square $ | 2 |
| Purple | $ 5^0 \times 5^\square $ | 9 |
| Brown | $ 25^2 \times 5^\square $ | 5 |
| Yellow | $ \dfrac{5^6 \times 5^\square}{5^2} $ | 5 |
| Reddish Orange | $ (5^\square)^{3/2} $ | 6 |
| Light Green | $ (5^\square)^9 $ | 1 |

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🎯 Final Answer:

All boxes should be filled as follows:

- Red: 9
- Cyan: 4
- Blue: 3
- Green: 3
- Orange: 2
- Purple: 9
- Brown: 5
- Yellow: 5
- Reddish Orange: 6
- Light Green: 1

Each expression now evaluates to $ 5^9 $, satisfying the condition. ✔