Calculate (-3)^4: Evaluating Negative Numbers Raised to Powers

Q: Why is (-3)^4 positive but (-3)^3 would be negative?

It's all about even vs odd exponents! With even exponents like 4, you multiply an even number of negatives, which always gives positive. With odd exponents like 3, you have an odd number of negatives, so the result stays negative.

Q: What's the difference between (-3)^4 and -3^4?

Huge difference! $(-3)^4$ means multiply -3 by itself 4 times = 81. But $-3^4$ means find 3^4 first, then make it negative = -81. Parentheses matter!

Q: How do I remember the sign rules for exponents?

Think of it like this: negative × negative = positive. So with even exponents, you're pairing up negatives to make positives. With odd exponents, there's always one negative left over!

Q: Can I use a calculator for problems like this?

Yes, but be careful with parentheses! Make sure to enter (-3)^4 not -3^4. Some calculators need you to use the negative button, not the minus button.

Q: Why do we multiply step by step instead of all at once?

Breaking it down helps you see the pattern and avoid mistakes! $(-3) \times (-3) = 9$, then $9 \times (-3) = -27$, then $-27 \times (-3) = 81$. Each step shows how the signs change.

Exponent Rules with Negative Base Numbers

$(-3)^4=$

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Step-by-step video solution

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00:00 Solve

00:04 First let's calculate the sign

00:07 First let's calculate the sign

00:11 Now let's calculate the exponent

00:18 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

$(-3)^4=$

Step-by-step solution

To solve the problem $(-3)^4$ , we'll perform the following computations step-by-step:

Step 1: Consider $(-3) \times (-3)$ . This equals $9$ because multiplying two negative numbers results in a positive number.
Step 2: Use the result from Step 1 and multiply it by $(-3)$ : $9 \times (-3) = -27$ .
Step 3: Finally, multiply the result from Step 2 by $(-3)$ : $-27 \times (-3) = 81$ .

The properties of exponents ensure that $(-3)^4$ results in a positive number because the exponent is even.

Therefore, the solution to the problem is $\textbf{81}$ . The correct answer corresponds to choice 3.

Final Answer

$81$

Key Points to Remember

Essential concepts to master this topic

Rule: Even exponents make negative bases become positive
Technique: Calculate step by step: (-3) × (-3) = 9, then 9 × (-3) × (-3) = 81
Check: Count multiplications: 4 negatives means 2 pairs, so result is positive ✓

Common Mistakes

Avoid these frequent errors

Forgetting parentheses and calculating -3^4 instead of (-3)^4
Don't calculate -3^4 = -(3^4) = -81! Without parentheses, you're finding the negative of 3^4, not raising -3 to the 4th power. Always include parentheses around negative bases before applying exponents.

Practice Quiz

Test your knowledge with interactive questions

Solve the following expression:

$$ $$ (-8)^2= $$

FAQ

Everything you need to know about this question

Why is (-3)^4 positive but (-3)^3 would be negative?

It's all about even vs odd exponents! With even exponents like 4, you multiply an even number of negatives, which always gives positive. With odd exponents like 3, you have an odd number of negatives, so the result stays negative.

What's the difference between (-3)^4 and -3^4?

Huge difference! $(-3)^4$ means multiply -3 by itself 4 times = 81. But $-3^4$ means find 3^4 first, then make it negative = -81. Parentheses matter!

How do I remember the sign rules for exponents?

Think of it like this: negative × negative = positive. So with even exponents, you're pairing up negatives to make positives. With odd exponents, there's always one negative left over!

Can I use a calculator for problems like this?

Yes, but be careful with parentheses! Make sure to enter (-3)^4 not -3^4. Some calculators need you to use the negative button, not the minus button.

Why do we multiply step by step instead of all at once?

Breaking it down helps you see the pattern and avoid mistakes! $(-3) \times (-3) = 9$ , then $9 \times (-3) = -27$ , then $-27 \times (-3) = 81$ . Each step shows how the signs change.

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