Problem: Simplify each problem and express the answer in scientific notation.
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Step-by-Step Solutions:
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1) \( (7 \times 10^6) \times (2 \times 10^8) \)
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Step 1: Multiply the coefficients: \( 7 \times 2 = 14 \).
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Step 2: Add the exponents of \( 10 \): \( 10^6 \times 10^8 = 10^{6+8} = 10^{14} \).
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Step 3: Combine the results: \( 14 \times 10^{14} \).
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Step 4: Convert to proper scientific notation: \( 1.4 \times 10^{15} \).
Answer: \( \boxed{1.4 \times 10^{15}} \)
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2) \( (9 \times 10^7) + (5 \times 10^7) \)
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Step 1: Since the powers of 10 are the same, add the coefficients: \( 9 + 5 = 14 \).
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Step 2: Keep the power of 10 as it is: \( 10^7 \).
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Step 3: Combine the results: \( 14 \times 10^7 \).
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Step 4: Convert to proper scientific notation: \( 1.4 \times 10^8 \).
Answer: \( \boxed{1.4 \times 10^8} \)
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3) \( (3 \times 10^6) - (7 \times 10^6) \)
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Step 1: Since the powers of 10 are the same, subtract the coefficients: \( 3 - 7 = -4 \).
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Step 2: Keep the power of 10 as it is: \( 10^6 \).
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Step 3: Combine the results: \( -4 \times 10^6 \).
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Step 4: This is already in scientific notation.
Answer: \( \boxed{-4 \times 10^6} \)
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4) \( \frac{5 \times 10^8}{4 \times 10^3} \)
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Step 1: Divide the coefficients: \( \frac{5}{4} = 1.25 \).
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Step 2: Subtract the exponents of \( 10 \): \( 10^8 \div 10^3 = 10^{8-3} = 10^5 \).
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Step 3: Combine the results: \( 1.25 \times 10^5 \).
Answer: \( \boxed{1.25 \times 10^5} \)
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5) \( \frac{2 \times 10^4}{8 \times 10^7} \)
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Step 1: Divide the coefficients: \( \frac{2}{8} = 0.25 \).
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Step 2: Subtract the exponents of \( 10 \): \( 10^4 \div 10^7 = 10^{4-7} = 10^{-3} \).
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Step 3: Combine the results: \( 0.25 \times 10^{-3} \).
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Step 4: Convert to proper scientific notation: \( 2.5 \times 10^{-4} \).
Answer: \( \boxed{2.5 \times 10^{-4}} \)
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6) \( (2 \times 10^3) - (4 \times 10^2) \)
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Step 1: Rewrite \( 4 \times 10^2 \) with the same exponent as \( 2 \times 10^3 \):
\[
4 \times 10^2 = 0.4 \times 10^3
\]
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Step 2: Subtract the coefficients: \( 2 - 0.4 = 1.6 \).
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Step 3: Keep the power of 10 as it is: \( 10^3 \).
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Step 4: Combine the results: \( 1.6 \times 10^3 \).
Answer: \( \boxed{1.6 \times 10^3} \)
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7) \( (6 \times 10^5) \div (1 \times 10^3) \)
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Step 1: Divide the coefficients: \( \frac{6}{1} = 6 \).
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Step 2: Subtract the exponents of \( 10 \): \( 10^5 \div 10^3 = 10^{5-3} = 10^2 \).
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Step 3: Combine the results: \( 6 \times 10^2 \).
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Step 4: This is already in scientific notation.
Answer: \( \boxed{6 \times 10^2} \)
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8) \( (8 \times 10^3) \times (6 \times 10^1) \)
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Step 1: Multiply the coefficients: \( 8 \times 6 = 48 \).
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Step 2: Add the exponents of \( 10 \): \( 10^3 \times 10^1 = 10^{3+1} = 10^4 \).
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Step 3: Combine the results: \( 48 \times 10^4 \).
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Step 4: Convert to proper scientific notation: \( 4.8 \times 10^5 \).
Answer: \( \boxed{4.8 \times 10^5} \)
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9) \( (4 \times 10^6) \times (9 \times 10^8) \)
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Step 1: Multiply the coefficients: \( 4 \times 9 = 36 \).
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Step 2: Add the exponents of \( 10 \): \( 10^6 \times 10^8 = 10^{6+8} = 10^{14} \).
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Step 3: Combine the results: \( 36 \times 10^{14} \).
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Step 4: Convert to proper scientific notation: \( 3.6 \times 10^{15} \).
Answer: \( \boxed{3.6 \times 10^{15}} \)
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10) \( (3 \times 10^9) + (7 \times 10^7) \)
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Step 1: Rewrite \( 7 \times 10^7 \) with the same exponent as \( 3 \times 10^9 \):
\[
7 \times 10^7 = 0.07 \times 10^9
\]
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Step 2: Add the coefficients: \( 3 + 0.07 = 3.07 \).
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Step 3: Keep the power of 10 as it is: \( 10^9 \).
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Step 4: Combine the results: \( 3.07 \times 10^9 \).
Answer: \( \boxed{3.07 \times 10^9} \)
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Final Answers:
1. \( \boxed{1.4 \times 10^{15}} \)
2. \( \boxed{1.4 \times 10^8} \)
3. \( \boxed{-4 \times 10^6} \)
4. \( \boxed{1.25 \times 10^5} \)
5. \( \boxed{2.5 \times 10^{-4}} \)
6. \( \boxed{1.6 \times 10^3} \)
7. \( \boxed{6 \times 10^2} \)
8. \( \boxed{4.8 \times 10^5} \)
9. \( \boxed{3.6 \times 10^{15}} \)
10. \( \boxed{3.07 \times 10^9} \)
Parent Tip: Review the logic above to help your child master the concept of math worksheet scientific notation.
