Complete the Expression: (3×4) Raised to Power (3x+1)

Q: Why can't I just multiply 3 × 4 = 12 first?

While $12^{3x+1}$ is mathematically correct, the question specifically asks for the expanded expression showing how the power rule works with individual factors.

Q: What's the difference between the power of a product and product of powers?

Power of a product: $(ab)^n = a^n \times b^n$Product of powers: $a^m \times a^n = a^{m+n}$These are different rules for different situations!

Q: Do both 3 and 4 get the same exponent?

Yes! When you have $(3 \times 4)^{3x+1}$, the entire exponent $3x+1$ applies to both the 3 and the 4 equally.

Q: What if the exponent was just a number like 2?

Same rule applies! $(3 \times 4)^2 = 3^2 \times 4^2 = 9 \times 16 = 144$. The power of a product rule works with any exponent.

Q: Can I use this rule with more than two factors?

Absolutely! $(abc)^n = a^n \times b^n \times c^n$. The exponent distributes to every single factor inside the parentheses.

Power of Product Rule with Exponential Expressions

Insert the corresponding expression:

$\left(3\times4\right)^{3x+1}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:08 Let's simplify this problem together.

00:12 To open parentheses with multiplication and an exponent outside,

00:16 we raise each factor inside to that power.

00:20 Now, let's apply this formula to our exercise.

00:24 Notice that the exponent has both addition and power N.

00:28 So we raise each factor to this power, step by step.

00:32 And that's how we find the solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\left(3\times4\right)^{3x+1}=$

Step-by-step solution

To solve this problem, we will apply the power of a product rule.

Step 1: Identify the given expression $(3 \times 4)^{3x+1}$ .
Step 2: Apply the power of a product rule: $(ab)^n = a^n \times b^n$ .
Step 3: Rewrite the expression using the rule:

By applying $(3 \times 4)^{3x+1} = 3^{3x+1} \times 4^{3x+1}$ , we distribute the exponent to each base within the parentheses.

Therefore, the correct expression is $3^{3x+1} \times 4^{3x+1}$ .

Final Answer

$3^{3x+1}\times4^{3x+1}$

Key Points to Remember

Essential concepts to master this topic

Rule: $(ab)^n = a^n \times b^n$ distributes exponent to each base
Technique: $(3 \times 4)^{3x+1} = 3^{3x+1} \times 4^{3x+1}$ applies rule directly
Check: Each base gets the same exponent $3x+1$ from original expression ✓

Common Mistakes

Avoid these frequent errors

Computing the product first then applying exponent
Don't calculate $3 \times 4 = 12$ first to get $12^{3x+1}$ = missing the point! The question asks for the expanded form using individual bases. Always distribute the exponent to each factor separately using $(ab)^n = a^n \times b^n$ .

Practice Quiz

Test your knowledge with interactive questions

$$ (4^2)^3+(g^3)^4= $$

FAQ

Everything you need to know about this question

Why can't I just multiply 3 × 4 = 12 first?

While $12^{3x+1}$ is mathematically correct, the question specifically asks for the expanded expression showing how the power rule works with individual factors.

What's the difference between the power of a product and product of powers?

Power of a product: $(ab)^n = a^n \times b^n$
Product of powers: $a^m \times a^n = a^{m+n}$
These are different rules for different situations!

Do both 3 and 4 get the same exponent?

Yes! When you have $(3 \times 4)^{3x+1}$ , the entire exponent $3x+1$ applies to both the 3 and the 4 equally.

What if the exponent was just a number like 2?

Same rule applies! $(3 \times 4)^2 = 3^2 \times 4^2 = 9 \times 16 = 144$ . The power of a product rule works with any exponent.

Can I use this rule with more than two factors?

Absolutely! $(abc)^n = a^n \times b^n \times c^n$ . The exponent distributes to every single factor inside the parentheses.

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