Question

simplify each problem and express the answer in scientific notation.

1. ((1.6 \times 10^{7}) - (3.4 \times 10^{4}))
2. ((9 \times 10^{7}) + (5 \times 10^{9}))
3. ((9 \times 10^{6}) - (8 \times 10^{2}))
4. ((12 \times 10^{4}) - (11 \times 10^{0}))
5. ((16 \times 10^{3}) + (6 \times 10^{1}))
6. ((2.5 \times 10^{8}) - (1.8 \times 10^{7}))
7. ((6 \times 10^{4}) + (1 \times 10^{3}))
8. ((9 \times 10^{5}) - (7 \times 10^{2}))
9. ((4.3 \times 10^{3}) - (0.4 \times 10^{1}))
10. ((3 \times 10^{5}) + (7 \times 10^{7}))
11. ((2.9 \times 10^{2}) + (0.7 \times 10^{2}))
12. ((9.5 \times 10^{3}) + (4.2 \times 10^{5}))

Explanation:

1) Step1: Match exponents

$1.6 \times 10^7 = 1600 \times 10^4$

1) Step2: Subtract terms

$1600 \times 10^4 - 3.4 \times 10^4 = (1600 - 3.4) \times 10^4 = 1596.6 \times 10^4$

1) Step3: Convert to scientific notation

$1596.6 \times 10^4 = 1.5966 \times 10^7$

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2) Step1: Match exponents

$9 \times 10^7 = 0.09 \times 10^9$

2) Step2: Add terms

$0.09 \times 10^9 + 5 \times 10^9 = (0.09 + 5) \times 10^9 = 5.09 \times 10^9$

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3) Step1: Match exponents

$9 \times 10^6 = 90000 \times 10^2$

3) Step2: Subtract terms

$90000 \times 10^2 - 8 \times 10^2 = (90000 - 8) \times 10^2 = 89992 \times 10^2$

3) Step3: Convert to scientific notation

$89992 \times 10^2 = 8.9992 \times 10^6$

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4) Step1: Match exponents

$12 \times 10^4 = 120000 \times 10^0$

4) Step2: Subtract terms

$120000 \times 10^0 - 11 \times 10^0 = (120000 - 11) \times 10^0 = 119989$

4) Step3: Convert to scientific notation

$119989 = 1.19989 \times 10^5$

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5) Step1: Match exponents

$16 \times 10^3 = 1600 \times 10^1$

5) Step2: Add terms

$1600 \times 10^1 + 6 \times 10^1 = (1600 + 6) \times 10^1 = 1606 \times 10^1$

5) Step3: Convert to scientific notation

$1606 \times 10^1 = 1.606 \times 10^4$

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6) Step1: Match exponents

$1.8 \times 10^7 = 0.18 \times 10^8$

6) Step2: Subtract terms

$2.5 \times 10^8 - 0.18 \times 10^8 = (2.5 - 0.18) \times 10^8 = 2.32 \times 10^8$

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7) Step1: Match exponents

$1 \times 10^3 = 0.1 \times 10^4$

7) Step2: Add terms

$6 \times 10^4 + 0.1 \times 10^4 = (6 + 0.1) \times 10^4 = 6.1 \times 10^4$

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8) Step1: Match exponents

$9 \times 10^5 = 9000 \times 10^2$

8) Step2: Subtract terms

$9000 \times 10^2 - 7 \times 10^2 = (9000 - 7) \times 10^2 = 8993 \times 10^2$

8) Step3: Convert to scientific notation

$8993 \times 10^2 = 8.993 \times 10^5$

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9) Step1: Match exponents

$4.3 \times 10^3 = 430 \times 10^1$

9) Step2: Subtract terms

$430 \times 10^1 - 0.4 \times 10^1 = (430 - 0.4) \times 10^1 = 429.6 \times 10^1$

9) Step3: Convert to scientific notation

$429.6 \times 10^1 = 4.296 \times 10^3$

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10) Step1: Match exponents

$3 \times 10^5 = 0.03 \times 10^7$

10) Step2: Add terms

$0.03 \times 10^7 + 7 \times 10^7 = (0.03 + 7) \times 10^7 = 7.03 \times 10^7$

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11) Step1: Add coefficients directly

$(2.9 + 0.7) \times 10^2 = 3.6 \times 10^2$

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12) Step1: Match exponents

$9.5 \times 10^3 = 0.095 \times 10^5$

12) Step2: Add terms

$0.095 \times 10^5 + 4.2 \times 10^5 = (0.095 + 4.2) \times 10^5 = 4.295 \times 10^5$

Answer:

1. $1.5966 \times 10^7$
2. $5.09 \times 10^9$
3. $8.9992 \times 10^6$
4. $1.19989 \times 10^5$
5. $1.606 \times 10^4$
6. $2.32 \times 10^8$
7. $6.1 \times 10^4$
8. $8.993 \times 10^5$
9. $4.296 \times 10^3$
10. $7.03 \times 10^7$
11. $3.6 \times 10^2$
12. $4.295 \times 10^5$