Simplify the Expression: 2³ × 3⁸ × 3⁹ × 2⁶

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

The product rule for exponents states that $ a^m \times a^n = a^{m+n} $. Think of it this way: $ 2^3 = 2 \times 2 \times 2 $ and $ 2^6 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 $, so together you have 9 factors of 2!

Q: What if the bases are different like 2 and 3?

When bases are different, you cannot combine them using exponent rules. Keep them separate! So $ 2^9 \times 3^{17} $ stays exactly as it is - this is the final simplified form.

Q: Can I rearrange the terms before simplifying?

Absolutely! Multiplication is commutative, so you can rearrange terms to group same bases together. This makes it much easier to apply exponent rules correctly.

Q: How do I know when I'm completely done simplifying?

You're done when you have only one term for each different base. In this problem, you should have one term with base 2 and one term with base 3, with no way to combine them further.

Exponent Rules with Mixed Base Terms

Reduce the following equation:

$2^3\times3^8\times3^9\times2^6=$

❤️ 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 laws of exponents, the multiplication of exponents with equal bases (A)

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

00:11 Let's identify the equal bases

00:14 We'll apply this formula to our exercise, one operation at a time

00:19 We'll maintain the base and add together the exponents

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:

$2^3\times3^8\times3^9\times2^6=$

Step-by-step solution

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

Step 1: Identify terms with the same base.
Step 2: Apply exponent rules to combine terms with the same base.
Step 3: Rewrite the expression in its simplified form.

Now, let's work through each step:
Step 1: Identify terms with the same base:
The original expression $2^3 \times 3^8 \times 3^9 \times 2^6$ contains two bases: 2 and 3.

Step 2: Apply exponent rules:
For base 2: $2^3 \times 2^6 = 2^{3+6} = 2^9$ .
For base 3: $3^8 \times 3^9 = 3^{8+9} = 3^{17}$ .

Step 3: Rewrite the expression in simplified form:
The expression simplifies to $2^9 \times 3^{17}$ .

Therefore, the solution to the problem is $2^9 \times 3^{17}$ . Thus, choice 4 is correct.

Final Answer

$2^9\times3^{17}$

Key Points to Remember

Essential concepts to master this topic

Same Base Rule: When multiplying powers with same bases, add exponents
Grouping: Rearrange to group same bases: $2^3 \times 2^6 = 2^9$
Verify: Check that all terms with same base are combined: $2^9 \times 3^{17}$ ✓

Common Mistakes

Avoid these frequent errors

Multiplying exponents instead of adding them
Don't multiply exponents like $2^3 \times 2^6 = 2^{18}$ ! This gives completely wrong results because you're applying power rules incorrectly. Always add exponents when multiplying same bases: $2^3 \times 2^6 = 2^{3+6} = 2^9$ .

Practice Quiz

Test your knowledge with interactive questions

$(3\times4\times5)^4=$

FAQ

Everything you need to know about this question

Why do I add the exponents instead of multiplying them?

The product rule for exponents states that $a^m \times a^n = a^{m+n}$ . Think of it this way: $2^3 = 2 \times 2 \times 2$ and $2^6 = 2 \times 2 \times 2 \times 2 \times 2 \times 2$ , so together you have 9 factors of 2!

What if the bases are different like 2 and 3?

When bases are different, you cannot combine them using exponent rules. Keep them separate! So $2^9 \times 3^{17}$ stays exactly as it is - this is the final simplified form.

Can I rearrange the terms before simplifying?

Absolutely! Multiplication is commutative, so you can rearrange terms to group same bases together. This makes it much easier to apply exponent rules correctly.

How do I know when I'm completely done simplifying?

You're done when you have only one term for each different base. In this problem, you should have one term with base 2 and one term with base 3, with no way to combine them further.

What's the difference between this and adding exponents with the same base?

This is adding exponents with the same base! The key is recognizing that $2^3 \times 2^6$ becomes $2^{3+6} = 2^9$ , and $3^8 \times 3^9$ becomes $3^{8+9} = 3^{17}$ .

🌟 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