Evaluate the Expression: 7¹¹ × 3¹¹ × 8 Using Exponent Properties

Q: Why can't I combine all three numbers under the exponent 11?

Because 8 doesn't have an exponent of 11! The power of product rule only works when all terms have the same exponent. Here, only $ 7^{11} $ and $ 3^{11} $ can be combined.

Q: What if 8 was written as 8¹¹ instead of just 8?

Then you could combine all three! If the expression were $ 7^{11} \times 3^{11} \times 8^{11} $, it would equal $ (7 \times 3 \times 8)^{11} $.

Q: Can I use this rule backwards too?

Yes! You can expand $ (ab)^n $ into $ a^n \times b^n $. This works in both directions - it's the same mathematical property.

Q: How do I remember when to apply this rule?

Look for multiplication of terms with identical exponents. If you see something like $ x^5 \times y^5 $, you can combine them as $ (xy)^5 $.

Q: What's the next step after getting (7×3)¹¹×8?

You could simplify further by calculating $ 7 \times 3 = 21 $, giving you $ 21^{11} \times 8 $. But the expression $ (7 \times 3)^{11} \times 8 $ is already correctly simplified!

Power of Product Rule with Mixed Terms

Choose the expression that corresponds to the following:

$7^{11}\times3^{11}\times8=$

❤️ 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

00:11 Let's simplify the equation.

00:14 We'll use the power of multiplication rule.

00:18 It says every factor multiplied to the power of N.

00:22 Is the same as each factor to the power of N.

00:26 Let's apply this formula in our exercise.

00:30 First, find equal bases and group them with parentheses.

00:46 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

Choose the expression that corresponds to the following:

$7^{11}\times3^{11}\times8=$

Step-by-step solution

To answer the question, we need to apply the power of a product exponent rule. This rule states that when you have a product of numbers raised to the same power, you can combine them under one exponent. The formula is as follows:

$a^n \times b^n = (a\times b)^n$

In our problem, we are given:
$7^{11}\times3^{11}\times8$

We notice that $7^{11}$ and $3^{11}$ are raised to the same power, 11. Therefore, according to the power of a product rule, we can combine them:

Identify the terms with the same power: $7^{11}$ and $3^{11}$ .
Combine the terms under a single exponent: $(7 \times 3)^{11}$ .

This means:

$7^{11}\times3^{11} = (7\times3)^{11}$

Thus, our simplified expression now looks like this:

$(7\times3)^{11}\times8$

Final Answer

$\left(7\times3\right)^{11}\times8$

Key Points to Remember

Essential concepts to master this topic

Rule: $a^n \times b^n = (a \times b)^n$ for same exponents
Technique: Combine $7^{11} \times 3^{11} = (7 \times 3)^{11}$ first
Check: Only terms with identical exponents can be combined using this rule ✓

Common Mistakes

Avoid these frequent errors

Combining all terms under one exponent
Don't write $(7 \times 3 \times 8)^{11}$ = wrong answer! The number 8 has no exponent (it's 8¹), so it can't be combined with terms that have exponent 11. Always combine only terms with the same exponent.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why can't I combine all three numbers under the exponent 11?

Because 8 doesn't have an exponent of 11! The power of product rule only works when all terms have the same exponent. Here, only $7^{11}$ and $3^{11}$ can be combined.

What if 8 was written as 8¹¹ instead of just 8?

Then you could combine all three! If the expression were $7^{11} \times 3^{11} \times 8^{11}$ , it would equal $(7 \times 3 \times 8)^{11}$ .

Can I use this rule backwards too?

Yes! You can expand $(ab)^n$ into $a^n \times b^n$ . This works in both directions - it's the same mathematical property.

How do I remember when to apply this rule?

Look for multiplication of terms with identical exponents. If you see something like $x^5 \times y^5$ , you can combine them as $(xy)^5$ .

What's the next step after getting (7×3)¹¹×8?

You could simplify further by calculating $7 \times 3 = 21$ , giving you $21^{11} \times 8$ . But the expression $(7 \times 3)^{11} \times 8$ is already correctly simplified!

🌟 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