Complete the Expression: (9×7×4) Raised to Power a

Q: Why can't I just raise one number to the power and leave the others?

Because the power applies to the entire product, not individual pieces! Think of it like distributing: when you have $ (a\times b)^n $, the exponent n affects both a and b equally.

Q: What's the difference between (9×7×4)^a and 9×7×4^a?

Huge difference! $ (9\times7\times4)^a $ means the entire product is raised to power a, while $ 9\times7\times4^a $ means only the 4 is raised to power a.

Q: Does order matter when I write the final answer?

No! $ 9^a\times7^a\times4^a $ is the same as $ 4^a\times9^a\times7^a $. Multiplication is commutative, so you can write the factors in any order.

Q: Can I combine the exponents somehow?

No! You cannot combine $ 9^a\times7^a\times4^a $ into something like $ (9\times7\times4)^{3a} $. The power of a product rule only works in one direction: expanding outward.

Q: What if the exponent was a number instead of 'a'?

The same rule applies! For example: $ (9\times7\times4)^2 = 9^2\times7^2\times4^2 = 81\times49\times16 $. You'd still raise each factor to that specific power.

Power of Product with Multiple Factors

Insert the corresponding expression:

$\left(9\times7\times4\right)^a=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following expression

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

00:08 We raise each factor to the power

00:13 We'll apply this formula to our exercise

00:22 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:

$\left(9\times7\times4\right)^a=$

Step-by-step solution

To solve this problem, let's apply the rule for the power of a product, which states that when a product is raised to a power, each factor is raised to that power.

Step 1: Write down the expression: $(9 \times 7 \times 4)^a$ .
Step 2: Apply the power of a product rule: $(x \times y \times z)^n = x^n \times y^n \times z^n$ .
Step 3: Raise each factor in the expression to the power $a$ .
Step 4: Simplify the individual expressions: $9^a \times 7^a \times 4^a$ .

Thus, the expression $(9 \times 7 \times 4)^a$ simplifies to $9^a \times 7^a \times 4^a$ .

The correct answer is $9^a \times 7^a \times 4^a$ , which corresponds to choice 2.

Final Answer

$9^a\times7^a\times4^a$

Key Points to Remember

Essential concepts to master this topic

Rule: When raising a product to a power, raise each factor separately
Technique: $(9\times7\times4)^a = 9^a\times7^a\times4^a$
Check: Verify each factor has the exponent a: count 9^a, 7^a, 4^a ✓

Common Mistakes

Avoid these frequent errors

Only raising some factors to the power
Don't apply the exponent to just one or two factors like $9^a\times7\times4^a$ = wrong result! This violates the power of a product rule and creates an unequal expression. Always raise every single factor in the product to the given power.

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 raise one number to the power and leave the others?

Because the power applies to the entire product, not individual pieces! Think of it like distributing: when you have $(a\times b)^n$ , the exponent n affects both a and b equally.

What's the difference between (9×7×4)^a and 9×7×4^a?

Huge difference! $(9\times7\times4)^a$ means the entire product is raised to power a, while $9\times7\times4^a$ means only the 4 is raised to power a.

Does order matter when I write the final answer?

No! $9^a\times7^a\times4^a$ is the same as $4^a\times9^a\times7^a$ . Multiplication is commutative, so you can write the factors in any order.

Can I combine the exponents somehow?

No! You cannot combine $9^a\times7^a\times4^a$ into something like $(9\times7\times4)^{3a}$ . The power of a product rule only works in one direction: expanding outward.

What if the exponent was a number instead of 'a'?

The same rule applies! For example: $(9\times7\times4)^2 = 9^2\times7^2\times4^2 = 81\times49\times16$ . You'd still raise each factor to that specific power.

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