Question

What is the missing exponent that would make this equation true?
k − superscript5⋅ k superscript2 / kx = (k0) − superscript2
(1 point)
Responses

x = 2
x = 2

x = -5
x = -5

x = -3
x = -3

x = 0

Answer 1

To solve for the missing exponent \( x \) in the equation

\[
\frac{k^{-5} \cdot k^{2}}{k^{x}} = k^{-2}
\]

we first simplify the left side of the equation. We can combine the exponents of \( k \) in the numerator:

\[
k^{-5} \cdot k^{2} = k^{-5 + 2} = k^{-3}
\]

Now our equation becomes:

\[
\frac{k^{-3}}{k^{x}} = k^{-2}
\]

Using the property of exponents \( \frac{a^{m}}{a^{n}} = a^{m-n} \), we can simplify the left-hand side:

\[
k^{-3 - x} = k^{-2}
\]

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

\[
-3 - x = -2
\]

Now we solve for \( x \):

\[
-x = -2 + 3
\]

\[
-x = 1
\]

\[
x = -1
\]

However, none of the options include \( x = -1 \). Let's double-check our steps:

1. Combine exponents on the left to get \( k^{-3} \).
2. Divide by \( k^x \) to get \( k^{-3 - x} \).
3. Set that equal to \( k^{-2} \).
4. Solve for \( x \) correctly.

So the correct analysis would indicate that none of the answers provided match \( x = -1\). Therefore, based on our calculation:

The answer **should be** \( x = -1\), which means the options provided were incorrect.

If required to pick the closest option given, the selected responses do not provide a correct answer given the derived condition \( x = -1\).