Simplify 3^-4 × 3^-2: Negative Exponents Multiplication Problem

Q: Why do I add the exponents instead of multiplying them?

The multiplication rule for exponents states: $ a^m \times a^n = a^{m+n} $. When you multiply powers with the same base, you add the exponents together.

Q: What happens when I add two negative numbers?

When adding negative numbers, you get a more negative result! Think of it as: -4 + (-2) = -4 - 2 = -6. The answer becomes more negative, not positive.

Q: Is 3^(-6) the final answer or do I need to simplify further?

For this problem, $ 3^{-6} $ is the simplified form requested. You could convert it to $ \frac{1}{3^6} $ or $ \frac{1}{729} $, but the exponential form is usually preferred.

Q: How do I remember the rule for multiplying powers?

Think of it as: "Same base, add the powers!" The base (3) stays the same, and you just add what's in the exponent positions: -4 + (-2) = -6.

Q: What if the exponents were positive instead?

The rule stays the same! $ 3^4 \times 3^2 = 3^{4+2} = 3^6 $. Whether the exponents are positive or negative, you still add them when multiplying powers with the same base.

Negative Exponents with Multiplication Rules

Simplify the following equation:

$3^{-4}\times3^{-2}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:09 Let's simplify this problem step by step.

00:12 Remember the rule for exponents: when multiplying with the same base, A,

00:18 the exponents add up. So, it's A to the power of N plus M.

00:24 Let's use this formula in our exercise.

00:28 Pay attention, we're adding negative exponents now.

00:32 When you multiply a positive by a negative, it becomes negative. So, we subtract.

00:39 And there you have it! That's the solution.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Simplify the following equation:

$3^{-4}\times3^{-2}=$

Final Answer

$3^{-4-2}$

Key Points to Remember

Essential concepts to master this topic

Rule: When multiplying powers with same base, add the exponents
Technique: $3^{-4} \times 3^{-2} = 3^{-4+(-2)} = 3^{-6}$
Check: Negative plus negative gives more negative: -4 + (-2) = -6 ✓

Common Mistakes

Avoid these frequent errors

Adding exponents incorrectly with negative signs
Don't ignore the negative signs and add -4 + (-2) = 2! This gives 3² instead of the correct answer. The negatives don't cancel out when adding. Always treat negative exponents carefully: -4 + (-2) = -6.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do I add the exponents instead of multiplying them?

The multiplication rule for exponents states: $a^m \times a^n = a^{m+n}$ . When you multiply powers with the same base, you add the exponents together.

What happens when I add two negative numbers?

When adding negative numbers, you get a more negative result! Think of it as: -4 + (-2) = -4 - 2 = -6. The answer becomes more negative, not positive.

Is 3^(-6) the final answer or do I need to simplify further?

For this problem, $3^{-6}$ is the simplified form requested. You could convert it to $\frac{1}{3^6}$ or $\frac{1}{729}$ , but the exponential form is usually preferred.

How do I remember the rule for multiplying powers?

Think of it as: "Same base, add the powers!" The base (3) stays the same, and you just add what's in the exponent positions: -4 + (-2) = -6.

What if the exponents were positive instead?

The rule stays the same! $3^4 \times 3^2 = 3^{4+2} = 3^6$ . Whether the exponents are positive or negative, you still add them when multiplying powers with the same base.

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