Answer :
Certainly! Let's break down each part of the problem step by step to find the solutions.
### Part (a)
[tex]\[ 2.3 \times 10^5 + 1.9 \times 10^5 \][/tex]
Step-by-step solution:
1. Recognize that both terms are powers of 10 to the same exponent.
2. Simply add the coefficients (2.3 and 1.9):
[tex]\[ 2.3 + 1.9 = 4.2 \][/tex]
3. Attach the power of 10:
[tex]\[ 4.2 \times 10^5 \][/tex]
4. Converting to standard notation:
[tex]\[ 4.2 \times 10^5 = 420000.0 \][/tex]
Thus, the solution is [tex]\( 420000.0 \)[/tex].
### Part (b)
[tex]\[ 4.5 \times 10^8 + 6.8 \times 10^7 \][/tex]
Step-by-step solution:
1. Convert [tex]\( 6.8 \times 10^7 \)[/tex] to the same exponent of 10:
[tex]\[ 6.8 \times 10^7 = 0.68 \times 10^8 \][/tex]
2. Add the coefficients:
[tex]\[ 4.5 + 0.68 = 5.18 \][/tex]
3. Attach the power of 10:
[tex]\[ 5.18 \times 10^8 \][/tex]
4. Converting to standard notation:
[tex]\[ 5.18 \times 10^8 = 518000000.0 \][/tex]
Thus, the solution is [tex]\( 518000000.0 \)[/tex].
### Part (c)
[tex]\[ 4.2 \times 10^6 + 1.3 \times 10^4 \][/tex]
Step-by-step solution:
1. Convert [tex]\( 1.3 \times 10^4 \)[/tex] to the same exponent of 10:
[tex]\[ 1.3 \times 10^4 = 0.013 \times 10^6 \][/tex]
2. Add the coefficients:
[tex]\[ 4.2 + 0.013 = 4.213 \][/tex]
3. Attach the power of 10:
[tex]\[ 4.213 \times 10^6 \][/tex]
4. Converting to standard notation:
[tex]\[ 4.213 \times 10^6 = 4213000.0 \][/tex]
Thus, the solution is [tex]\( 4213000.0 \)[/tex].
### Part (d)
[tex]\[ 5.8 \times 10^7 + 3.4 \times 10^3 \][/tex]
Step-by-step solution:
1. Convert [tex]\( 3.4 \times 10^3 \)[/tex] to the same exponent of 10:
[tex]\[ 3.4 \times 10^3 = 0.000034 \times 10^7 \][/tex]
2. Add the coefficients:
[tex]\[ 5.8 + 0.000034 = 5.800034 \][/tex]
3. Attach the power of 10:
[tex]\[ 5.800034 \times 10^7 \][/tex]
4. Converting to standard notation:
[tex]\[ 5.800034 \times 10^7 = 58003400.0 \][/tex]
Thus, the solution is [tex]\( 58003400.0 \)[/tex].
### Part (e)
[tex]\[ 8.9 \times 10^4 + 1.1 \times 10^{-2} \][/tex]
Step-by-step solution:
1. Convert [tex]\( 1.1 \times 10^{-2} \)[/tex] to the same exponent of 10:
[tex]\[ 1.1 \times 10^{-2} = 0.000011 \times 10^4 \][/tex]
2. Add the coefficients:
[tex]\[ 8.9 + 0.000011 = 8.900011 \][/tex]
3. Attach the power of 10:
[tex]\[ 8.900011 \times 10^4 \][/tex]
4. Converting to standard notation:
[tex]\[ 8.900011 \times 10^4 = 89000.011 \][/tex]
Thus, the solution is [tex]\( 89000.011 \)[/tex].
### Part (f)
[tex]\[ 7.8 \times 10^5 + 1.0 \times 10^{-6} \][/tex]
Step-by-step solution:
1. Convert [tex]\( 1.0 \times 10^{-6} \)[/tex] to the same exponent of 10:
[tex]\[ 1.0 \times 10^{-6} = 0.000001 \times 10^5 \][/tex]
2. Add the coefficients:
[tex]\[ 7.8 + 0.000001 = 7.800001 \][/tex]
3. Attach the power of 10:
[tex]\[ 7.800001 \times 10^5 \][/tex]
4. Converting to standard notation:
[tex]\[ 7.800001 \times 10^5 = 780000.000001 \][/tex]
Thus, the solution is [tex]\( 780000.000001 \)[/tex].
These are the detailed, step-by-step solutions for each part of the given problem.
### Part (a)
[tex]\[ 2.3 \times 10^5 + 1.9 \times 10^5 \][/tex]
Step-by-step solution:
1. Recognize that both terms are powers of 10 to the same exponent.
2. Simply add the coefficients (2.3 and 1.9):
[tex]\[ 2.3 + 1.9 = 4.2 \][/tex]
3. Attach the power of 10:
[tex]\[ 4.2 \times 10^5 \][/tex]
4. Converting to standard notation:
[tex]\[ 4.2 \times 10^5 = 420000.0 \][/tex]
Thus, the solution is [tex]\( 420000.0 \)[/tex].
### Part (b)
[tex]\[ 4.5 \times 10^8 + 6.8 \times 10^7 \][/tex]
Step-by-step solution:
1. Convert [tex]\( 6.8 \times 10^7 \)[/tex] to the same exponent of 10:
[tex]\[ 6.8 \times 10^7 = 0.68 \times 10^8 \][/tex]
2. Add the coefficients:
[tex]\[ 4.5 + 0.68 = 5.18 \][/tex]
3. Attach the power of 10:
[tex]\[ 5.18 \times 10^8 \][/tex]
4. Converting to standard notation:
[tex]\[ 5.18 \times 10^8 = 518000000.0 \][/tex]
Thus, the solution is [tex]\( 518000000.0 \)[/tex].
### Part (c)
[tex]\[ 4.2 \times 10^6 + 1.3 \times 10^4 \][/tex]
Step-by-step solution:
1. Convert [tex]\( 1.3 \times 10^4 \)[/tex] to the same exponent of 10:
[tex]\[ 1.3 \times 10^4 = 0.013 \times 10^6 \][/tex]
2. Add the coefficients:
[tex]\[ 4.2 + 0.013 = 4.213 \][/tex]
3. Attach the power of 10:
[tex]\[ 4.213 \times 10^6 \][/tex]
4. Converting to standard notation:
[tex]\[ 4.213 \times 10^6 = 4213000.0 \][/tex]
Thus, the solution is [tex]\( 4213000.0 \)[/tex].
### Part (d)
[tex]\[ 5.8 \times 10^7 + 3.4 \times 10^3 \][/tex]
Step-by-step solution:
1. Convert [tex]\( 3.4 \times 10^3 \)[/tex] to the same exponent of 10:
[tex]\[ 3.4 \times 10^3 = 0.000034 \times 10^7 \][/tex]
2. Add the coefficients:
[tex]\[ 5.8 + 0.000034 = 5.800034 \][/tex]
3. Attach the power of 10:
[tex]\[ 5.800034 \times 10^7 \][/tex]
4. Converting to standard notation:
[tex]\[ 5.800034 \times 10^7 = 58003400.0 \][/tex]
Thus, the solution is [tex]\( 58003400.0 \)[/tex].
### Part (e)
[tex]\[ 8.9 \times 10^4 + 1.1 \times 10^{-2} \][/tex]
Step-by-step solution:
1. Convert [tex]\( 1.1 \times 10^{-2} \)[/tex] to the same exponent of 10:
[tex]\[ 1.1 \times 10^{-2} = 0.000011 \times 10^4 \][/tex]
2. Add the coefficients:
[tex]\[ 8.9 + 0.000011 = 8.900011 \][/tex]
3. Attach the power of 10:
[tex]\[ 8.900011 \times 10^4 \][/tex]
4. Converting to standard notation:
[tex]\[ 8.900011 \times 10^4 = 89000.011 \][/tex]
Thus, the solution is [tex]\( 89000.011 \)[/tex].
### Part (f)
[tex]\[ 7.8 \times 10^5 + 1.0 \times 10^{-6} \][/tex]
Step-by-step solution:
1. Convert [tex]\( 1.0 \times 10^{-6} \)[/tex] to the same exponent of 10:
[tex]\[ 1.0 \times 10^{-6} = 0.000001 \times 10^5 \][/tex]
2. Add the coefficients:
[tex]\[ 7.8 + 0.000001 = 7.800001 \][/tex]
3. Attach the power of 10:
[tex]\[ 7.800001 \times 10^5 \][/tex]
4. Converting to standard notation:
[tex]\[ 7.800001 \times 10^5 = 780000.000001 \][/tex]
Thus, the solution is [tex]\( 780000.000001 \)[/tex].
These are the detailed, step-by-step solutions for each part of the given problem.
