To solve the equation [tex]\(5 \times 10^c + 5 \times 10^d = 50,500\)[/tex], we can follow these steps:
1. Start simplifying the equation: Notice that each term in the equation [tex]\(5 \times 10^c + 5 \times 10^d = 50,500\)[/tex] has a common factor of 5. We can divide both sides of the equation by 5 to simplify it:
[tex]\[
\frac{5 \times 10^c + 5 \times 10^d}{5} = \frac{50,500}{5}
\][/tex]
This reduces to:
[tex]\[
10^c + 10^d = 10,100
\][/tex]
2. Analyze possible values for [tex]\(c\)[/tex] and [tex]\(d\)[/tex]: We need to find two powers of 10 that add up to 10,100. To find appropriate values for [tex]\(c\)[/tex] and [tex]\(d\)[/tex], let’s try different combinations:
- Consider [tex]\(10^c = 10,000\)[/tex] (which corresponds to [tex]\(c = 4\)[/tex]), compare it with:
[tex]\[
10^4 + 10^d = 10,100 \quad \text{gives} \quad 10^d = 10,100 - 10,000 = 100
\][/tex]
Hence, [tex]\(10^d = 100\)[/tex] which corresponds to [tex]\(d = 2\)[/tex].
3. Verify the solution: Plug back the values [tex]\(c = 4\)[/tex] and [tex]\(d = 2\)[/tex] into the original equation to check if they satisfy the condition:
[tex]\[
5 \times 10^4 + 5 \times 10^2 = 5 \times 10,000 + 5 \times 100 = 50,000 + 500 = 50,500
\][/tex]
The left-hand side of the equation equals the right-hand side, confirming that our chosen values are correct.
Thus, the values for [tex]\(c\)[/tex] and [tex]\(d\)[/tex] that satisfy the equation are [tex]\(\boxed{4}\)[/tex] and [tex]\(\boxed{2}\)[/tex].
