Compare (-2)⁷ and 2⁸: Exploring Negative Base Exponents

To solve this problem, we'll follow these steps:

• Step 1: Calculate $(-2)^7$
• Step 2: Calculate $-2^8$
• Step 3: Compare the two results

Now, let's work through each step:

Step 1: Calculate $(-2)^7$.

Using the power of negative numbers rule, $(-2)^7$ is a negative number because 7 is odd. We perform the calculation:

$(-2)^7 = - (2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2) = -128$.

Step 2: Calculate $-2^8$.

Here, $2^8$ is positive since 8 is an even number:

$2^8 = 256$.

But, note the negative sign in front: $-2^8 = -(2^8) = -256$.

Step 3: Compare $(-2)^7$ and $-2^8$:

We have $(-2)^7 = -128$ and $-2^8 = -256$. The comparison shows:

$-128 > -256$.

Therefore, the correct comparison is $(-2)^7 > -2^8$.

By following the steps and verifying calculations, we conclude that $$ $(-2)^7 > -2^8$.