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Sagot :
Sure, let's solve each part step-by-step.
### Part (a)
Given expression:
[tex]\[ \frac{1.5 \times 10^4 \times 4.5 \times 10^3}{2.5 \times 10^2} \][/tex]
First, calculate the numerator:
[tex]\[ 1.5 \times 10^4 \times 4.5 \times 10^3 = (1.5 \times 4.5) \times (10^4 \times 10^3) = 6.75 \times 10^7 \][/tex]
Next, calculate the denominator:
[tex]\[ 2.5 \times 10^2 = 2.5 \times 10^2 \][/tex]
Now, perform the division:
[tex]\[ \frac{6.75 \times 10^7}{2.5 \times 10^2} = \frac{6.75}{2.5} \times \frac{10^7}{10^2} = 2.7 \times 10^5 = 270000.0 \][/tex]
### Part (b)
Given expression:
[tex]\[ \frac{6.4 \times 10^{-5} \times 3.6 \times 10^9}{1.6 \times 10^{10} \times 1.8 \times 10^{-3}} \][/tex]
First, calculate the numerator:
[tex]\[ 6.4 \times 10^{-5} \times 3.6 \times 10^9 = (6.4 \times 3.6) \times (10^{-5} \times 10^9) = 23.04 \times 10^4 = 2.304 \times 10^5 \][/tex]
Next, calculate the denominator:
[tex]\[ 1.6 \times 10^{10} \times 1.8 \times 10^{-3} = (1.6 \times 1.8) \times (10^{10} \times 10^{-3}) = 2.88 \times 10^7 \][/tex]
Now, perform the division:
[tex]\[ \frac{2.304 \times 10^5}{2.88 \times 10^7} = \frac{2.304}{2.88} \times 10^{-2} = 0.8 \times 10^{-2} = 0.008 \][/tex]
### Part (c)
Given expression:
[tex]\[ \frac{6.7 \times 10^{-11} \times 2.5 \times 10^{20} \times 3.6 \times 10^{15}}{(3.35 \times 10^6)^2} \][/tex]
First, calculate the numerator:
[tex]\[ 6.7 \times 10^{-11} \times 2.5 \times 10^{20} \times 3.6 \times 10^{15} = (6.7 \times 2.5 \times 3.6) \times (10^{-11} \times 10^{20} \times 10^{15}) = 60.3 \times 10^{24} \][/tex]
Next, calculate the denominator:
[tex]\[ (3.35 \times 10^6)^2 = (3.35^2) \times (10^6)^2 = 11.2225 \times 10^{12} \][/tex]
Now, perform the division:
[tex]\[ \frac{60.3 \times 10^{24}}{11.2225 \times 10^{12}} = \frac{60.3}{11.2225} \times 10^{12} \approx 5.37 \times 10^{12} = 5373134328358.209 \][/tex]
### Part (d)
Given expression:
[tex]\[ \frac{7.8 \times 10^7 + 6.5 \times 10^7}{1.3 \times 10^3} \][/tex]
First, calculate the numerator:
[tex]\[ 7.8 \times 10^7 + 6.5 \times 10^7 = (7.8 + 6.5) \times 10^7 = 14.3 \times 10^7 \][/tex]
Next, calculate the denominator:
[tex]\[ 1.3 \times 10^3 \][/tex]
Now, perform the division:
[tex]\[ \frac{14.3 \times 10^7}{1.3 \times 10^3} = \frac{14.3}{1.3} \times 10^{7-3} = 11 \times 10^4 = 110000.0 \][/tex]
### Part (e)
Given expression:
[tex]\[ 3.2 \times 10^3 + 4.8 \times 10^4 \][/tex]
Perform the addition directly:
[tex]\[ 3.2 \times 10^3 + 4.8 \times 10^4 = 3200 + 48000 = 51200.0 \][/tex]
### Part (f)
Given expression:
[tex]\[ \frac{9.6 \times 10^6 - 7.2 \times 10^5}{2.4 \times 10^3} \][/tex]
First, calculate the numerator:
[tex]\[ 9.6 \times 10^6 - 7.2 \times 10^5 = 9600000 - 720000 = 8880000 \][/tex]
Next, calculate the denominator:
[tex]\[ 2.4 \times 10^3 = 2400 \][/tex]
Now, perform the division:
[tex]\[ \frac{8880000}{2400} = 3700.0 \][/tex]
So the results for each part are:
[tex]\[ (a) \ 270000.0, \quad (b) \ 0.008, \quad (c) \ 5373134328358.209, \quad (d) \ 110000.0, \quad (e) \ 51200.0, \quad (f) \ 3700.0 \][/tex]
### Part (a)
Given expression:
[tex]\[ \frac{1.5 \times 10^4 \times 4.5 \times 10^3}{2.5 \times 10^2} \][/tex]
First, calculate the numerator:
[tex]\[ 1.5 \times 10^4 \times 4.5 \times 10^3 = (1.5 \times 4.5) \times (10^4 \times 10^3) = 6.75 \times 10^7 \][/tex]
Next, calculate the denominator:
[tex]\[ 2.5 \times 10^2 = 2.5 \times 10^2 \][/tex]
Now, perform the division:
[tex]\[ \frac{6.75 \times 10^7}{2.5 \times 10^2} = \frac{6.75}{2.5} \times \frac{10^7}{10^2} = 2.7 \times 10^5 = 270000.0 \][/tex]
### Part (b)
Given expression:
[tex]\[ \frac{6.4 \times 10^{-5} \times 3.6 \times 10^9}{1.6 \times 10^{10} \times 1.8 \times 10^{-3}} \][/tex]
First, calculate the numerator:
[tex]\[ 6.4 \times 10^{-5} \times 3.6 \times 10^9 = (6.4 \times 3.6) \times (10^{-5} \times 10^9) = 23.04 \times 10^4 = 2.304 \times 10^5 \][/tex]
Next, calculate the denominator:
[tex]\[ 1.6 \times 10^{10} \times 1.8 \times 10^{-3} = (1.6 \times 1.8) \times (10^{10} \times 10^{-3}) = 2.88 \times 10^7 \][/tex]
Now, perform the division:
[tex]\[ \frac{2.304 \times 10^5}{2.88 \times 10^7} = \frac{2.304}{2.88} \times 10^{-2} = 0.8 \times 10^{-2} = 0.008 \][/tex]
### Part (c)
Given expression:
[tex]\[ \frac{6.7 \times 10^{-11} \times 2.5 \times 10^{20} \times 3.6 \times 10^{15}}{(3.35 \times 10^6)^2} \][/tex]
First, calculate the numerator:
[tex]\[ 6.7 \times 10^{-11} \times 2.5 \times 10^{20} \times 3.6 \times 10^{15} = (6.7 \times 2.5 \times 3.6) \times (10^{-11} \times 10^{20} \times 10^{15}) = 60.3 \times 10^{24} \][/tex]
Next, calculate the denominator:
[tex]\[ (3.35 \times 10^6)^2 = (3.35^2) \times (10^6)^2 = 11.2225 \times 10^{12} \][/tex]
Now, perform the division:
[tex]\[ \frac{60.3 \times 10^{24}}{11.2225 \times 10^{12}} = \frac{60.3}{11.2225} \times 10^{12} \approx 5.37 \times 10^{12} = 5373134328358.209 \][/tex]
### Part (d)
Given expression:
[tex]\[ \frac{7.8 \times 10^7 + 6.5 \times 10^7}{1.3 \times 10^3} \][/tex]
First, calculate the numerator:
[tex]\[ 7.8 \times 10^7 + 6.5 \times 10^7 = (7.8 + 6.5) \times 10^7 = 14.3 \times 10^7 \][/tex]
Next, calculate the denominator:
[tex]\[ 1.3 \times 10^3 \][/tex]
Now, perform the division:
[tex]\[ \frac{14.3 \times 10^7}{1.3 \times 10^3} = \frac{14.3}{1.3} \times 10^{7-3} = 11 \times 10^4 = 110000.0 \][/tex]
### Part (e)
Given expression:
[tex]\[ 3.2 \times 10^3 + 4.8 \times 10^4 \][/tex]
Perform the addition directly:
[tex]\[ 3.2 \times 10^3 + 4.8 \times 10^4 = 3200 + 48000 = 51200.0 \][/tex]
### Part (f)
Given expression:
[tex]\[ \frac{9.6 \times 10^6 - 7.2 \times 10^5}{2.4 \times 10^3} \][/tex]
First, calculate the numerator:
[tex]\[ 9.6 \times 10^6 - 7.2 \times 10^5 = 9600000 - 720000 = 8880000 \][/tex]
Next, calculate the denominator:
[tex]\[ 2.4 \times 10^3 = 2400 \][/tex]
Now, perform the division:
[tex]\[ \frac{8880000}{2400} = 3700.0 \][/tex]
So the results for each part are:
[tex]\[ (a) \ 270000.0, \quad (b) \ 0.008, \quad (c) \ 5373134328358.209, \quad (d) \ 110000.0, \quad (e) \ 51200.0, \quad (f) \ 3700.0 \][/tex]
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