Complete the Expression: (9×2)^(3x) Mathematical Pattern

Q: Why can't I just multiply 9×2 first to get 18?

You could simplify to $ 18^{3x} $, but the question asks for the expanded form showing each base separately. Both $ 18^{3x} $ and $ 9^{3x} \times 2^{3x} $ are mathematically equal!

Q: Does the order matter in the final answer?

No! Since multiplication is commutative, $ 9^{3x} \times 2^{3x} $ equals $ 2^{3x} \times 9^{3x} $. Both orders are correct.

Q: What if the exponent was just a number instead of 3x?

The same rule applies! For example, $ (9×2)^5 = 9^5 \times 2^5 $. The power of a product rule works with any exponent, whether it's a number, variable, or expression.

Q: How do I remember this rule?

Think: "Power spreads to everyone!" When you have $ (ab)^n $, the exponent n gets distributed to both a and b, just like handing out the same thing to each person.

Q: Can I use this rule backwards?

Yes! If you see $ 9^{3x} \times 2^{3x} $, you can factor it as $ (9×2)^{3x} $. This reverse application is called factoring out common exponents.

Power of Products with Variable Exponents

Insert the corresponding expression:

$\left(9\times2\right)^{3x}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:04 In order to open parentheses with a multiplication operation and an outside exponent

00:08 Raise each factor to the power (N)

00:11 We will apply this formula to our exercise

00:14 Note that all the factors in the multiplication operation are raised to the same exponent (N)

00:17 Therefore, we will raise each factor to this power (N)

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:

$\left(9\times2\right)^{3x}=$

Step-by-step solution

To solve this problem, we'll apply the power of a product rule for exponents:

Identify that $(9 \times 2)^{3x}$ is a product of two numbers $9$ and $2$ raised to the same power $3x$ .
Apply the power of a product rule: $(ab)^n = a^n \times b^n$ .
In our case, $a = 9$ , $b = 2$ , and $n = 3x$ .
Thus, $(9 \times 2)^{3x} = 9^{3x} \times 2^{3x}$ .

Therefore, the expression $(9 \times 2)^{3x}$ simplifies to $9^{3x} \times 2^{3x}$ .

Final Answer

$9^{3x}\times2^{3x}$

Key Points to Remember

Essential concepts to master this topic

Rule: Apply power to each factor: $(ab)^n = a^n \times b^n$
Technique: Distribute 3x to both 9 and 2: $(9×2)^{3x} = 9^{3x} \times 2^{3x}$
Check: Verify using numerical values: if x=1, then $(18)^3 = 18^3$ ✓

Common Mistakes

Avoid these frequent errors

Applying exponent to only one factor
Don't write $(9×2)^{3x} = 9×2^{3x}$ = wrong distribution! This ignores the power rule and gives incorrect results. Always apply the exponent to each factor separately: $9^{3x} \times 2^{3x}$ .

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 9×2 first to get 18?

You could simplify to $18^{3x}$ , but the question asks for the expanded form showing each base separately. Both $18^{3x}$ and $9^{3x} \times 2^{3x}$ are mathematically equal!

Does the order matter in the final answer?

No! Since multiplication is commutative, $9^{3x} \times 2^{3x}$ equals $2^{3x} \times 9^{3x}$ . Both orders are correct.

What if the exponent was just a number instead of 3x?

The same rule applies! For example, $(9×2)^5 = 9^5 \times 2^5$ . The power of a product rule works with any exponent, whether it's a number, variable, or expression.

How do I remember this rule?

Think: "Power spreads to everyone!" When you have $(ab)^n$ , the exponent n gets distributed to both a and b, just like handing out the same thing to each person.

Can I use this rule backwards?

Yes! If you see $9^{3x} \times 2^{3x}$ , you can factor it as $(9×2)^{3x}$ . This reverse application is called factoring out common exponents.

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