Express 64 as Different Mathematical Representations

Q: What's the difference between (-8)² and -(8)²?

$ (-8)^2 $ means the entire negative number is squared: $ (-8) \times (-8) = 64 $. But $ -(8)^2 $ means square 8 first, then apply the negative: $ -(64) = -64 $.

Q: Why does a negative number squared become positive?

When you multiply two negative numbers, you get a positive result! Think of it as: negative × negative = positive. So $ (-8) \times (-8) = +64 $.

Q: Are there other negative numbers that square to 64?

Yes! Both 8 and -8 square to 64. We say 64 has two square roots: +8 (the principal root) and -8. But $ (-8)^2 = 64 $ is the correct negative base representation.

Q: How do I remember the parentheses rule?

Remember: parentheses include everything inside when applying the exponent. Without parentheses around (-8), only the 8 gets squared, and the negative sign stays outside!

Q: What if I need to find other ways to write 64?

64 has many representations! $ 8^2 $, $ (-8)^2 $, $ 4^3 $, $ 2^6 $. The key is understanding which form matches what you're asked to find.

Negative Base Exponents with Perfect Squares

$64=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:03 Let's break down the power.

00:06 We'll split the power into multiplication, including its sign.

00:10 This method works well.

00:14 Now let's split the power into multiplication, without the sign.

00:19 And that's how we solve this problem!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$64=$

Step-by-step solution

To solve this problem and express 64 as a power involving a negative number, we will follow these steps:

Step 1: Recognize that we need to represent 64 using a base number squared. Since 64 is a perfect square, let's consider negative integers whose square equals 64.
Step 2: The principal positive square root of 64 is 8. However, we are tasked with finding a negative number such that its square is 64.
Step 3: If we have a negative integer, $(-8)$ , and square it, we have: $(-8)^2 = (-8) \times (-8) = 64$ .
Step 4: Compare this with the expression $-(8)^2$ , which results in $-64$ because the square applies only to 8, and the negative sign flips the result.

Therefore, the correct expression representing 64 with a negative base is $(-8)^2$ , and among the answer choices provided, choice 1 is the correct one.

Final Answer

$(-8)^2$

Key Points to Remember

Essential concepts to master this topic

Rule: Parentheses around negative base include sign in calculation
Technique: $(-8)^2 = (-8) \times (-8) = 64$
Check: Verify negative times negative equals positive result ✓

Common Mistakes

Avoid these frequent errors

Confusing parentheses placement with negative signs
Don't write $-(8)^2$ when you mean $(-8)^2$ = wrong sign! The first gives -64 because only 8 is squared, then negated. Always put parentheses around the entire negative number before applying the exponent.

Practice Quiz

Test your knowledge with interactive questions

$$ (-2)^7= $$

FAQ

Everything you need to know about this question

What's the difference between (-8)² and -(8)²?

$(-8)^2$ means the entire negative number is squared: $(-8) \times (-8) = 64$ . But $-(8)^2$ means square 8 first, then apply the negative: $-(64) = -64$ .

Why does a negative number squared become positive?

When you multiply two negative numbers, you get a positive result! Think of it as: negative × negative = positive. So $(-8) \times (-8) = +64$ .

Are there other negative numbers that square to 64?

Yes! Both 8 and -8 square to 64. We say 64 has two square roots: +8 (the principal root) and -8. But $(-8)^2 = 64$ is the correct negative base representation.

How do I remember the parentheses rule?

Remember: parentheses include everything inside when applying the exponent. Without parentheses around (-8), only the 8 gets squared, and the negative sign stays outside!

What if I need to find other ways to write 64?

64 has many representations! $8^2$ , $(-8)^2$ , $4^3$ , $2^6$ . The key is understanding which form matches what you're asked to find.

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