Answer :
To find the product of
[tex]$$8.2 \times 10^9 \quad \text{and} \quad 4.5 \times 10^{-5},$$[/tex]
we follow these steps:
1. Multiply the coefficients:
Multiply [tex]$8.2$[/tex] by [tex]$4.5$[/tex]:
[tex]$$
8.2 \times 4.5 = 36.9.
$$[/tex]
2. Combine the powers of 10:
Since the powers have the same base, add the exponents:
[tex]$$
10^9 \times 10^{-5} = 10^{9+(-5)} = 10^4.
$$[/tex]
At this point, the multiplication gives:
[tex]$$
36.9 \times 10^4.
$$[/tex]
3. Normalize the result:
In standard scientific notation, the coefficient must be between [tex]$1$[/tex] and [tex]$10$[/tex]. The number [tex]$36.9$[/tex] is not in this range. We can express [tex]$36.9$[/tex] as:
[tex]$$
36.9 = 3.69 \times 10^1.
$$[/tex]
Substitute this into the product:
[tex]$$
36.9 \times 10^4 = (3.69 \times 10^1) \times 10^4.
$$[/tex]
4. Combine the exponents:
Combine [tex]$10^1$[/tex] and [tex]$10^4$[/tex]:
[tex]$$
10^1 \times 10^4 = 10^{1+4} = 10^5.
$$[/tex]
Therefore, the product in standard scientific notation is:
[tex]$$
3.69 \times 10^5.
$$[/tex]
Thus, the final answer is:
[tex]$$
\boxed{3.69 \times 10^5}.
$$[/tex]
[tex]$$8.2 \times 10^9 \quad \text{and} \quad 4.5 \times 10^{-5},$$[/tex]
we follow these steps:
1. Multiply the coefficients:
Multiply [tex]$8.2$[/tex] by [tex]$4.5$[/tex]:
[tex]$$
8.2 \times 4.5 = 36.9.
$$[/tex]
2. Combine the powers of 10:
Since the powers have the same base, add the exponents:
[tex]$$
10^9 \times 10^{-5} = 10^{9+(-5)} = 10^4.
$$[/tex]
At this point, the multiplication gives:
[tex]$$
36.9 \times 10^4.
$$[/tex]
3. Normalize the result:
In standard scientific notation, the coefficient must be between [tex]$1$[/tex] and [tex]$10$[/tex]. The number [tex]$36.9$[/tex] is not in this range. We can express [tex]$36.9$[/tex] as:
[tex]$$
36.9 = 3.69 \times 10^1.
$$[/tex]
Substitute this into the product:
[tex]$$
36.9 \times 10^4 = (3.69 \times 10^1) \times 10^4.
$$[/tex]
4. Combine the exponents:
Combine [tex]$10^1$[/tex] and [tex]$10^4$[/tex]:
[tex]$$
10^1 \times 10^4 = 10^{1+4} = 10^5.
$$[/tex]
Therefore, the product in standard scientific notation is:
[tex]$$
3.69 \times 10^5.
$$[/tex]
Thus, the final answer is:
[tex]$$
\boxed{3.69 \times 10^5}.
$$[/tex]