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

Q: What's the difference between (-2)^7 and -2^7?

Parentheses matter! $ (-2)^7 $ means multiply -2 by itself 7 times = -128. But $ -2^7 $ means calculate $ 2^7 $ first, then make it negative = -128. They're the same here, but different for even exponents!

Q: Why is (-2)^7 negative but (-2)^8 would be positive?

It's all about odd vs even exponents! When you multiply an odd number of negative values, you get negative. With an even number of negatives, you get positive. Odd exponent = negative result, even exponent = positive result.

Q: How do I compare two negative numbers like -128 and -256?

Think of a number line! The number closer to zero is larger. Since -128 is closer to 0 than -256, we have $ -128 > -256 $. Remember: with negatives, smaller absolute value means larger number!

Q: What does -2^8 mean exactly?

$ -2^8 $ means $ -(2^8) $ by order of operations. First calculate $ 2^8 = 256 $, then apply the negative sign to get -256. The negative is not part of the base being raised to the 8th power.

Q: How can I remember the order of operations with exponents?

Use PEMDAS! Parentheses first, then Exponents, then Multiplication/Division. So $ -2^8 $ becomes $ -(2^8) $ because exponents come before the negative (multiplication by -1).

Negative Exponents with Order of Operations

Which is larger?

$(-2)^7⬜-2^8$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Place the appropriate sign

00:03 Let's calculate the exponent, the sign is not part of the exponent

00:06 Now let's calculate the sign of the second exponent

00:10 Odd exponent, therefore the sign remains negative

00:16 Let's calculate the exponent and compare

00:26 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Which is larger?

$(-2)^7⬜-2^8$

Step-by-step solution

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 $(-2)^7 > -2^8$ .

Final Answer

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Key Points to Remember

Essential concepts to master this topic

Rule: Parentheses determine if negative applies to base or result
Technique: (-2)^7 = -128, but -2^8 = -(2^8) = -256
Check: Compare -128 > -256 on number line: closer to zero wins ✓

Common Mistakes

Avoid these frequent errors

Confusing (-2)^7 with -2^7
Don't treat (-2)^7 the same as -2^7 = they give different results! (-2)^7 means the negative is part of the base, while -2^7 means negate after calculating. Always check if parentheses include the negative sign.

Practice Quiz

Test your knowledge with interactive questions

$$ (-2)^7= $$

FAQ

Everything you need to know about this question

What's the difference between (-2)^7 and -2^7?

Parentheses matter! $(-2)^7$ means multiply -2 by itself 7 times = -128. But $-2^7$ means calculate $2^7$ first, then make it negative = -128. They're the same here, but different for even exponents!

Why is (-2)^7 negative but (-2)^8 would be positive?

It's all about odd vs even exponents! When you multiply an odd number of negative values, you get negative. With an even number of negatives, you get positive. Odd exponent = negative result, even exponent = positive result.

How do I compare two negative numbers like -128 and -256?

Think of a number line! The number closer to zero is larger. Since -128 is closer to 0 than -256, we have $-128 > -256$ . Remember: with negatives, smaller absolute value means larger number!

What does -2^8 mean exactly?

$-2^8$ means $-(2^8)$ by order of operations. First calculate $2^8 = 256$ , then apply the negative sign to get -256. The negative is not part of the base being raised to the 8th power.

How can I remember the order of operations with exponents?

Use PEMDAS! Parentheses first, then Exponents, then Multiplication/Division. So $-2^8$ becomes $-(2^8)$ because exponents come before the negative (multiplication by -1).

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