Calculate (-2)^7: Evaluating the Seventh Power of a Negative Number

Q: Why is (-2)⁷ negative but (-2)⁶ would be positive?

It's all about odd vs even exponents! When you multiply a negative number an odd number of times (like 7), you get negative. When you multiply it an even number of times (like 6), the negatives cancel out and you get positive.

Q: What's the difference between (-2)⁷ and -2⁷?

Big difference! $ (-2)^7 = -128 $ because the parentheses include the negative sign in the base. But $ -2^7 = -(2^7) = -128 $ means "negative of 2 to the 7th power." In this case, both equal -128, but be careful with even exponents!

Q: How can I remember the sign rule for exponents?

Think of it like this: negative × negative = positive. So if you pair up the negatives, any leftover negative (odd exponent) makes the result negative. Even exponents = all negatives pair up = positive result.

Q: Is there a faster way to calculate 2⁷ without multiplying step by step?

Yes! You can use powers of 2 patterns: $ 2^1=2, 2^2=4, 2^3=8, 2^4=16, 2^5=32, 2^6=64, 2^7=128 $. Or use the fact that $ 2^7 = 2^6 \times 2 = 64 \times 2 = 128 $.

Negative Number Exponents with Odd Powers

$(-2)^7=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:04 Let's solve the problem together.

00:07 First, we'll figure out the sign.

00:10 Since the power is odd, the sign will be negative.

00:20 Now, let's calculate the power itself.

00:26 And that's the answer to our question!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$(-2)^7=$

Step-by-step solution

To solve for $(-2)^7$ , follow these steps:

Step 1: Identify the base and the exponent given in the expression, which are $-2$ and $7$ , respectively.
Step 2: Recognize that since the exponent is $7$ , which is an odd number, the result of the power will remain negative: $(-2)^7$ will be $- (2^7)$ .
Step 3: Compute $2^7$ . This involves multiplying $2$ by itself $7$ times:
$2 \times 2 = 4$
$4 \times 2 = 8$
$8 \times 2 = 16$
$16 \times 2 = 32$
$32 \times 2 = 64$
$64 \times 2 = 128$
Thus, $2^7 = 128$ .
Step 4: Apply the negative sign to the result of $2^7$ , resulting in $-128$ .

Therefore, the value of $(-2)^7$ is $-128$ .

Final Answer

$-128$

Key Points to Remember

Essential concepts to master this topic

Sign Rule: Odd exponents keep the negative sign unchanged
Technique: Calculate $2^7 = 128$ , then apply negative sign
Check: Count multiplications: (-2)×(-2)×(-2)×(-2)×(-2)×(-2)×(-2) = negative ✓

Common Mistakes

Avoid these frequent errors

Dropping the negative sign when calculating powers
Don't calculate $(-2)^7$ as just $2^7 = 128$ ! This ignores the crucial sign rule and gives a positive answer instead of negative. Always remember that odd exponents preserve the negative sign, making $(-2)^7 = -128$ .

Practice Quiz

Test your knowledge with interactive questions

$$ (-2)^7= $$

FAQ

Everything you need to know about this question

Why is (-2)⁷ negative but (-2)⁶ would be positive?

It's all about odd vs even exponents! When you multiply a negative number an odd number of times (like 7), you get negative. When you multiply it an even number of times (like 6), the negatives cancel out and you get positive.

Do I need to write out all the multiplications?

Not necessarily! Once you understand the pattern, you can use the shortcut: calculate the positive version first ( $2^7 = 128$ ), then apply the sign rule. Odd exponent = negative result.

What's the difference between (-2)⁷ and -2⁷?

Big difference! $(-2)^7 = -128$ because the parentheses include the negative sign in the base. But $-2^7 = -(2^7) = -128$ means "negative of 2 to the 7th power." In this case, both equal -128, but be careful with even exponents!

How can I remember the sign rule for exponents?

Think of it like this: negative × negative = positive. So if you pair up the negatives, any leftover negative (odd exponent) makes the result negative. Even exponents = all negatives pair up = positive result.

Is there a faster way to calculate 2⁷ without multiplying step by step?

Yes! You can use powers of 2 patterns: $2^1=2, 2^2=4, 2^3=8, 2^4=16, 2^5=32, 2^6=64, 2^7=128$ . Or use the fact that $2^7 = 2^6 \times 2 = 64 \times 2 = 128$ .

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