Simplify the Expression: 1/(2×3)^(-5) Using Negative Exponents

Q: Why does dividing by a negative exponent make it positive?

Think of it as "flipping twice"! The negative exponent already flipped $ (2×3)^{-5} $ to the denominator. When you divide by it (put it under 1), you flip it back to the numerator as $ (2×3)^5 $.

Q: Do I need to calculate 2×3 first?

Not necessarily! The problem asks for the equivalent expression, not the numerical answer. $ (2×3)^5 $ is the correct form, though you could also write it as $ 6^5 $.

Q: What's the difference between negative exponents and negative numbers?

Great question! A negative exponent like $ 3^{-2} $ means $ \frac{1}{3^2} $. A negative base like $ (-3)^2 $ means the number itself is negative. They're completely different concepts!

Q: How can I remember the negative exponent rule?

Remember: "Negative exponents mean flip!" $ a^{-n} = \frac{1}{a^n} $ and $ \frac{1}{a^{-n}} = a^n $. When you see 1 divided by a negative exponent, just flip it to make the exponent positive!

Negative Exponents with Reciprocal Rules

Insert the corresponding expression:

$\frac{1}{\left(2\times3\right)^{-5}}=$

❤️ 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 Apply the exponent laws in order to simplify the negative exponents

00:06 Convert to the reciprocal number and raise it to the power of (-1)

00:10 We'll apply this formula to our exercise

00:13 Convert to the reciprocal number (1 divided by the number)

00:17 Raise to the power of (-1)

00:20 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\frac{1}{\left(2\times3\right)^{-5}}=$

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Identify the given expression: $\frac{1}{(2 \times 3)^{-5}}$ .
Step 2: Apply the negative exponent rule. This rule states that $\frac{1}{a^{-n}} = a^{n}$ .
Step 3: Simplify the expression.

Now, let's work through each step:

Step 1: The problem gives us the expression $\frac{1}{(2 \times 3)^{-5}}$ .

Step 2: Apply the negative exponent rule:

Given $\frac{1}{(2 \times 3)^{-5}}$ , we use the formula for negative exponents, which means:

$(a \times b)^{-n} = \frac{1}{(a \times b)^{n}}$

We can rewrite $\frac{1}{(2 \times 3)^{-5}}$ as $(2 \times 3)^{5}$ .

There is no need for further simplification, as the problem asks only for the equivalent expression.

Therefore, the expression $\frac{1}{(2 \times 3)^{-5}}$ simplifies to $(2 \times 3)^{5}$ .

The correct answer is $(2 \times 3)^{5}$ .

Final Answer

$\left(2\times3\right)^5$

Key Points to Remember

Essential concepts to master this topic

Rule: $\frac{1}{a^{-n}} = a^n$ converts negative exponents to positive
Technique: $\frac{1}{(2×3)^{-5}}$ becomes $(2×3)^5$ directly
Check: Verify using $a^{-n} = \frac{1}{a^n}$ backwards: $(6)^{-5} = \frac{1}{6^5}$ ✓

Common Mistakes

Avoid these frequent errors

Adding negative signs when dealing with negative exponents
Don't think $\frac{1}{(2×3)^{-5}}$ equals $-(2×3)^5$ = negative result! The negative exponent only affects the position (numerator vs denominator), not the sign of the answer. Always apply the reciprocal rule: $\frac{1}{a^{-n}} = a^n$ keeps the positive result.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why does dividing by a negative exponent make it positive?

Think of it as "flipping twice"! The negative exponent already flipped $(2×3)^{-5}$ to the denominator. When you divide by it (put it under 1), you flip it back to the numerator as $(2×3)^5$ .

Do I need to calculate 2×3 first?

Not necessarily! The problem asks for the equivalent expression, not the numerical answer. $(2×3)^5$ is the correct form, though you could also write it as $6^5$ .

What's the difference between negative exponents and negative numbers?

Great question! A negative exponent like $3^{-2}$ means $\frac{1}{3^2}$ . A negative base like $(-3)^2$ means the number itself is negative. They're completely different concepts!

Why isn't the answer 2^(-5) × 3^(-5)?

That would be if we distributed the negative exponent first: $(2×3)^{-5} = 2^{-5} × 3^{-5}$ . But the question asks us to simplify $\frac{1}{(2×3)^{-5}}$ , so we use the reciprocal rule directly on the whole expression.

How can I remember the negative exponent rule?

Remember: "Negative exponents mean flip!" $a^{-n} = \frac{1}{a^n}$ and $\frac{1}{a^{-n}} = a^n$ . When you see 1 divided by a negative exponent, just flip it to make the exponent positive!

🌟 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