What is the missing exponent that would make this equation true?

Question

(k^-5 time k^2)/(k^x) = (k^0)^-2

A: x = 0
B: x = -5
C: x = -3
D: x = 2

Answer 1

To solve this equation, we can simplify each side separately and then equate the two expressions.

Starting with the left side of the equation:
(k^(-5) * k^2) / k^x

When we multiply terms with the same base, we add their exponents. Therefore:
k^(-5 + 2) / k^x = k^(-3) / k^x

Now, we need to apply the exponent rules for division. When we divide terms with the same base, we subtract the exponents. Therefore:
k^(-3) / k^x = k^(-3 - x)

Moving on to the right side of the equation:
(k^0)^(-2)

Any term raised to the 0th power is equal to 1. Therefore:
1^(-2) = 1

Now that we have the simplified expressions, we can equate them and solve for x:
k^(-3 - x) = 1

Since any term raised to the power 0 is equal to 1, the exponent on the right side must be 0. Therefore:
-3 - x = 0

Simplifying this equation, we find:
x = -3

Therefore, the missing exponent that would make the equation true is x = -3.

The correct option is C: x = -3.