1. **Problem:** Multiply the given numbers by powers of 10. Formula: Multiplying by $10^n$ shifts the decimal point $n$ places to the right if $n$ is positive, or $|n|$ places to the left if $n$ is negative. Rules: $10^0=1$, multiplying by $10^0$ leaves the number unchanged. Calculations: a) $13 \times 10^7 = 130000000$ b) $7.8 \times 10^3 = 7800$ c) $24 \times 10 = 240$ d) $8.55 \times 10^4 = 85500$ e) $6.5 \times 10^1 = 65$ f) $0.08 \times 10^5 = 8000$ g) $17 \times 10^0 = 17$ h) $8 \times 10^{-1} = 0.8$ i) $8.5 \times 10^{-2} = 0.085$ j) $4500 \times 10^{-4} = 0.45$ k) $32 \times 10^{-1} = 3.2$ l) $125 \times 10^{-2} = 1.25$ 2. **Problem:** Divide the given numbers by powers of 10. Formula: Dividing by $10^n$ shifts the decimal point $n$ places to the left if $n$ is positive, or $|n|$ places to the right if $n$ is negative. Calculations: a) $27 \div 10 = 2.7$ b) $450 \div 10^3 = 0.45$ c) $36 \div 10^2 = 0.36$ d) $170 \div 10^4 = 0.017$ e) $0.8 \div 10^1 = 0.08$ f) $2480 \div 10^6 = 0.00248$ g) $9 \div 10^2 = 0.09$ h) $0.25 \div 10^1 = 0.025$ i) $18 \div 10^2 = 0.18$ j) $4.76 \div 10^4 = 0.000476$ k) $0.07 \div 10^1 = 0.007$ l) $0.085 \div 10^{-1} = 0.85$ 3. **Problem:** Decode the secret message using the code table and calculations. Calculations and letters: $0.6 \times 10^2 = 60 \Rightarrow S$ $0.06 \div 10^{-1} = 0.06 \times 10^1 = 0.6 \Rightarrow A$ $60 \times 10^{-1} = 6 \Rightarrow L$ $600 \div 10^{-1} = 600 \times 10^1 = 6000 \Rightarrow R$ $60 \times 10^{-1} = 6 \Rightarrow H$ $33 \div 10^0 = 33 \Rightarrow E$ $0.33 \div 10^1 = 0.033 \Rightarrow W$ $300 \times 10^{-1} = 30 \Rightarrow N$ $300 \div 10^1 = 30 \Rightarrow C$ $3300 \times 10^{-2} = 33 \Rightarrow E$ $0.3 \div 10^1 = 0.03 \Rightarrow T$ $300 \div 10^{-1} = 300 \times 10^1 = 3000 \Rightarrow D$ Secret message: "SALE RH EWNCE TD" (letters above numbers in the table) 4. **Problem:** Find missing powers in spider diagrams so all outer values equal the center. a) Center: $8 \times 10^4$ Check each outer value: - $0.8 \times 10^5 = 8 \times 10^4$ - $8 \times 10^3 = 0.8 \times 10^4$ - $80 \times 10^2 = 8 \times 10^3$ - $0.08 \times 10^6 = 8 \times 10^4$ - $0.008 \times 10^7 = 8 \times 10^4$ - $800 \times 10^1 = 8 \times 10^3$ - $8 = 8 \times 10^0$ b) Center: $32 \times 10^4$ Check each outer value: - $0.32 \times 10^5 = 32 \times 10^4$ - $3.2 \times 10^5 = 32 \times 10^4$ - $320 \times 10^3 = 32 \times 10^4$ - $32 \times 10^3 = 3.2 \times 10^4$ - $3.2 \times 10^4 = 3.2 \times 10^4$ - $0.32 = 3.2 \times 10^{-1}$ - $0.32 = 3.2 \times 10^{-1}$ 5. a) Multiply by powers of 10: $i) 4 \times 10^2 = 400$ $iii) 4 \times 10^0 = 4$ $iv) 4 \times 10^{-1} = 0.4$ $v) 4 \times 10^{-2} = 0.04$ $vi) 4 \times 10^{-3} = 0.004$ b) Multiplying by $10^4$ makes the number larger because it shifts the decimal 4 places right. Multiplying by $10^{-2}$ makes the number smaller because it shifts the decimal 2 places left. 6. a) Multiply by powers of 10: $i) 12 \times 10^1 = 120$ $ii) 12 \times 10^0 = 12$ $iii) 12 \times 10^{-1} = 1.2$ $iv) 12 \times 10^{-1} = 1.2$ $v) 12 \times 10^{-2} = 0.12$ $vi) 12 \times 10^{-3} = 0.012$ b) Dividing by $10^{-1}$ is the same as multiplying by 10, so the answer is larger. Dividing by $10^{-3}$ is multiplying by 1000, so the answer is even larger. Therefore, dividing by $10^{-3}$ gives a larger result than dividing by $10^{-1}$. Remember: $10^0=1$, $10^{-1}=0.1$.