Question

Which expression simplified would leave you with a negative exponent when simplified in exponent form?

options:

2^−12 over 2^−6

2^5 ⋅ 2^−2

−2(−3)^2

2^3−2^4

Answer 1

To determine which expression simplifies to include a negative exponent, let's analyze each option one by one.

1. **Expression: \( \frac{2^{-12}}{2^{-6}} \)**

Apply the quotient rule of exponents:
\[
\frac{a^m}{a^n} = a^{m-n}
\]
\[
\frac{2^{-12}}{2^{-6}} = 2^{-12 - (-6)} = 2^{-12 + 6} = 2^{-6}
\]
This expression simplifies to \( 2^{-6} \), which is indeed a negative exponent.

2. **Expression: \( 2^5 \cdot 2^{-2} \)**

Apply the product rule of exponents:
\[
a^m \cdot a^n = a^{m+n}
\]
\[
2^5 \cdot 2^{-2} = 2^{5 + (-2)} = 2^{5 - 2} = 2^3
\]
This simplifies to \( 2^3 \), which is a positive exponent.

3. **Expression: \( -2(-3)^2 \)**

Simplifying the expression:
\[
-2(-3)^2 = -2 \cdot 9 = -18
\]
This does not involve exponents at all and results in a negative value but not a negative exponent.

4. **Expression: \( 2^3 - 2^4 \)**

This expression does not involve any exponents being multiplied or divided:
\[
2^3 - 2^4 = 8 - 16 = -8
\]
While the calculation results in a negative number, it does not produce a negative exponent.

After reviewing all options, the correct expression that results in a negative exponent when simplified is:

**\( \frac{2^{-12}}{2^{-6}} = 2^{-6} \)**.