In the following equation, what are possible values for c and d?

To solve the equation ( 4^c cdot 4^d = 4^9 ), we can use the properties of exponents.

When we multiply two expressions with the same base, we can add their exponents:

[

4^c cdot 4^d = 4^{c+d}

]

This means we can rewrite the equation as:

[

4^{c+d} = 4^9

]

Since the bases are the same, we can set the exponents equal to each other:

[

c + d = 9

]

Now, (c) and (d) can be any pair of numbers that add up to 9. For example:

– If (c = 0), then (d = 9).

– If (c = 1), then (d = 8).

– If (c = 4), then (d = 5).

– If (c = 9), then (d = 0).

So, there are many possible values for (c) and (d) as long as they satisfy the equation (c + d = 9).

In summary:

– (c) can be any value, and (d) will be (9 – c).

– (d) can be any value, and (c) will be (9 – d