Simplify 5^-3 × 5^-4: Negative Exponent Multiplication

Q: Why do we add negative exponents instead of subtracting them?

Because we're adding the exponents, not the signs! When you add (-3) + (-4), you get -7. Think of it as moving 3 steps backward, then 4 more steps backward = 7 steps backward total.

Q: What does a negative exponent actually mean?

A negative exponent means "one over" that positive power. So $ 5^{-7} = \frac{1}{5^7} $. It's the reciprocal of the positive exponent!

Q: Can I just ignore the negative signs and add normally?

No! The negative signs are part of the exponents. You must include them when adding: (-3) + (-4) = -7, not 3 + 4 = 7. The sign matters for your final answer!

Q: How is this different from dividing exponents?

When multiplying same bases, you add exponents. When dividing same bases, you subtract exponents. Since we're multiplying here ($ \times $), we add!

Q: What if the bases were different, like 3 and 5?

You cannot combine different bases using exponent rules! $ 3^{-2} \times 5^{-4} $ stays as is - the bases must be the same to add exponents.

Exponent Rules with Negative Powers

Reduce the following equation:

$5^{-3}\times5^{-4}=$

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:03 According to the power laws, the multiplication of powers with equal bases (A)

00:07 equals the same base raised to the sum of the powers (N+M)

00:11 We will apply this formula to our exercise

00:15 Note that we are adding a negative factor

00:21 A positive x A negative is always negative, therefore we subtract as follows

00:26 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Reduce the following equation:

$5^{-3}\times5^{-4}=$

Step-by-step solution

To solve the problem of simplifying the expression $5^{-3} \times 5^{-4}$ , we will apply the exponent multiplication rule, which states that when multiplying powers with the same base, we simply add their exponents.

Step 1: Identify the expression and confirm both terms share the base of 5. They are $5^{-3}$ and $5^{-4}$ .
Step 2: Apply the exponent multiplication rule: $5^{-3} \times 5^{-4} = 5^{(-3) + (-4)}$ .
Step 3: Simplify the expression by adding the exponents: $-3 + (-4) = -7$ .

Therefore, the expression simplifies to $5^{-7}$ .

Given the possible choices, the correct answer is $5^{-7}$ , which corresponds to choice (1).

Thus, the solution to the problem is $5^{-7}$ .

Final Answer

$5^{-7}$

Key Points to Remember

Essential concepts to master this topic

Multiplication Rule: When multiplying same bases, add the exponents
Technique: $5^{-3} \times 5^{-4} = 5^{(-3) + (-4)} = 5^{-7}$
Check: Convert to fractions: $\frac{1}{5^3} \times \frac{1}{5^4} = \frac{1}{5^7} = 5^{-7}$ ✓

Common Mistakes

Avoid these frequent errors

Multiplying the exponents instead of adding them
Don't multiply -3 × -4 = 12 to get $5^{12}$ ! This confuses the power rule with the multiplication rule and gives completely wrong answers. Always add exponents when multiplying same bases: (-3) + (-4) = -7.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do we add negative exponents instead of subtracting them?

Because we're adding the exponents, not the signs! When you add (-3) + (-4), you get -7. Think of it as moving 3 steps backward, then 4 more steps backward = 7 steps backward total.

What does a negative exponent actually mean?

A negative exponent means "one over" that positive power. So $5^{-7} = \frac{1}{5^7}$ . It's the reciprocal of the positive exponent!

Can I just ignore the negative signs and add normally?

No! The negative signs are part of the exponents. You must include them when adding: (-3) + (-4) = -7, not 3 + 4 = 7. The sign matters for your final answer!

How is this different from dividing exponents?

When multiplying same bases, you add exponents. When dividing same bases, you subtract exponents. Since we're multiplying here ( $\times$ ), we add!

What if the bases were different, like 3 and 5?

You cannot combine different bases using exponent rules! $3^{-2} \times 5^{-4}$ stays as is - the bases must be the same to add exponents.

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Exponents Rules questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime