Question

Question
What is the missing exponent that would make this equation true?
k−5⋅ k2kx=(k0)−2
𝑘
−
5
⋅

𝑘
2
𝑘
𝑥
=
(
𝑘
0
)
−
2
(1 point)
Responses

x = 0
x = 0

x = -5
x = -5

x = -3
x = -3

x = 2

Answer 1

To solve the equation \( k^{-5} \cdot k^{2} k^{x} = (k^{0})^{-2} \), we can simplify both sides and find the value of \( x \).

1. **Simplify the left side:**
\[
k^{-5} \cdot k^{2} \cdot k^{x} = k^{-5 + 2 + x} = k^{-3 + x}
\]

2. **Simplify the right side:**
\[
(k^{0})^{-2} = 1^{-2} = 1 = k^{0}
\]

Since any non-zero number raised to the power of zero equals one, we have:
\[
(k^{0})^{-2} = k^{0}
\]

Now we can equate the exponents from both sides:
\[
-3 + x = 0
\]

3. **Solve for \( x \):**
\[
x = 3
\]

There appears to be no option listed for \( x = 3 \). Considering there may have been a misunderstanding in the options, please re-evaluate them, but based on this calculation, the missing exponent should be \( x = 3 \).

However, if we carefully re-evaluate the original equation with available options, it seems there is no match. The closest calculation must be recalibrated with existing options, or the input values should be rechecked.

If the options must be strictly adhered to, a review of misunderstandings or misprints in the prompt is recommended. Otherwise, from the proper calculation \( x \) should conclude as \( 3 \) instead of what is currently provided in responses.