Evaluate the Complex Expression: 3⁹×12⁴×6⁹×4⁹×4⁴×7⁹

Q: Why can't I just multiply all the bases together with one exponent?

Because the terms don't all have the same exponent! You have $ 12^4 $ and $ 4^4 $ mixed with terms that have exponent 9. You must group by matching exponents first.

Q: What is the power of product rule exactly?

The power of product rule states: $ a^n \times b^n = (a \times b)^n $. This only works when the exponents are the same. For example: $ 2^3 \times 5^3 = (2 \times 5)^3 = 10^3 $.

Q: How do I identify which terms to group together?

Look at the exponents, not the bases! Group $ 3^9, 6^9, 4^9, 7^9 $ together (all have exponent 9), and $ 12^4, 4^4 $ together (both have exponent 4).

Q: Can I calculate the actual numerical answer?

You could, but it would be an enormous number! The question asks for the simplified form using exponent rules, which is much more manageable: $ (3 \times 6 \times 4 \times 7)^9 \times (12 \times 4)^4 $.

Q: What if I see terms like 4⁹ and 4⁴ in the same problem?

Keep them separate! $ 4^9 $ goes with other 9th power terms, while $ 4^4 $ goes with other 4th power terms. Different exponents mean different groups.

Exponent Laws with Grouping Strategy

$3^9\times12^4\times6^9\times4^9\times4^4\times7^9=\text{ ?}$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:14 Let's simplify this problem together.

00:17 Remember, if a product is raised to the power of N, each factor inside is also raised to the power of N.

00:24 Our job is to find all numbers with the same exponents.

00:28 Next, let's apply this rule step by step to our exercise.

00:33 Find numbers with matching exponents. Then, apply the rule we discussed.

00:39 We carefully use the formula, and we solve it step by step.

00:47 And that's how we get our solution! Great job!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$3^9\times12^4\times6^9\times4^9\times4^4\times7^9=\text{ ?}$

Step-by-step solution

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

Step 1: Group terms with the same exponent to utilize the power of a product rule.
Step 2: Combine terms under each group using the power of a product property.
Step 3: Match the result to one of the multiple-choice options provided.

Now, let's work through each step:
Step 1: Observe the expression $3^9 \times 12^4 \times 6^9 \times 4^9 \times 4^4 \times 7^9$ . Group the terms with exponents of 9: $3^9, 6^9, 4^9,$ and $7^9$ . Group those with exponents of 4: $12^4$ and $4^4$ .

Step 2: For the terms with exponent 9, apply the power of a product rule: $(3 \times 6 \times 4 \times 7)^9$

For the terms with exponent 4, apply the power of a product rule: $(12 \times 4)^4$

Step 3: Combine these to form the expression: $(3 \times 6 \times 4 \times 7)^9 \times (12 \times 4)^4$

Therefore, the solution to the problem is $\left(3 \times 6 \times 4 \times 7\right)^9 \times \left(12 \times 4\right)^4$ . This corresponds to choice 3.

Final Answer

$\left(3\times6\times4\times7\right)^9\times\left(12\times4\right)^4$

Key Points to Remember

Essential concepts to master this topic

Rule: Group terms with same exponents to apply power of product law
Technique: $3^9 \times 6^9 \times 4^9 \times 7^9 = (3 \times 6 \times 4 \times 7)^9$
Check: Count exponents: four terms with 9, two terms with 4 gives final answer ✓

Common Mistakes

Avoid these frequent errors

Trying to apply exponent rules to all terms at once
Don't combine all terms as $(3 \times 12 \times 6 \times 4 \times 4 \times 7)^9$ = incorrect factorization! This ignores that some terms have different exponents (4 vs 9). Always group terms by their exponents first, then apply the power of product rule to each group separately.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why can't I just multiply all the bases together with one exponent?

Because the terms don't all have the same exponent! You have $12^4$ and $4^4$ mixed with terms that have exponent 9. You must group by matching exponents first.

What is the power of product rule exactly?

The power of product rule states: $a^n \times b^n = (a \times b)^n$ . This only works when the exponents are the same. For example: $2^3 \times 5^3 = (2 \times 5)^3 = 10^3$ .

How do I identify which terms to group together?

Look at the exponents, not the bases! Group $3^9, 6^9, 4^9, 7^9$ together (all have exponent 9), and $12^4, 4^4$ together (both have exponent 4).

Can I calculate the actual numerical answer?

You could, but it would be an enormous number! The question asks for the simplified form using exponent rules, which is much more manageable: $(3 \times 6 \times 4 \times 7)^9 \times (12 \times 4)^4$ .

What if I see terms like 4⁹ and 4⁴ in the same problem?

Keep them separate! $4^9$ goes with other 9th power terms, while $4^4$ goes with other 4th power terms. Different exponents mean different groups.

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