Calculate (9×6×8)^5: Fifth Power of a Product Expression

Q: Why can I rearrange the factors in any order?

Because multiplication is commutative! This means $ a \times b = b \times a $. So $ 9^5 \times 6^5 \times 8^5 $ equals $ 8^5 \times 6^5 \times 9^5 $ and any other arrangement.

Q: Do I need to calculate the actual numbers?

Not necessarily! The question asks for the expression, not the numerical value. Leaving it as $ 9^5 \times 6^5 \times 8^5 $ is perfectly correct and often preferred.

Q: What if there were more factors inside the parentheses?

The same rule applies! For $ (a \times b \times c \times d)^n $, you get $ a^n \times b^n \times c^n \times d^n $. Every factor gets raised to the power.

Q: Can I simplify before applying the exponent?

You could calculate $ 9 \times 6 \times 8 = 432 $ first to get $ 432^5 $, but that's usually harder to work with than separate factors.

Q: How do I remember this power rule?

Think of it as distributing the exponent! Just like you distribute multiplication over addition, you distribute exponents over multiplication inside parentheses.

Power of Products with Multiple Factors

Choose the expression that corresponds to the following:

$\left(9\times6\times8\right)^5=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:09 Let's simplify this problem together.

00:13 When you have multiplication with each factor raised to the same exponent, N.

00:18 You can raise each factor to that power, N.

00:22 Let's use this idea in our exercise, step by step.

00:31 Remember, in multiplication, changing the order of numbers doesn't change the result.

00:40 Now, we'll reorder the factors using our rule and solve the exercise.

01:02 Great job! That's the solution.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Choose the expression that corresponds to the following:

$\left(9\times6\times8\right)^5=$

Step-by-step solution

To solve the given problem, we'll utilize the power of a product rule. This rule states that:

$(a \times b \times c)^n = a^n \times b^n \times c^n$

Let's apply this rule to the expression $(9 \times 6 \times 8)^5$ :

$(9 \times 6 \times 8)^5 = 9^5 \times 6^5 \times 8^5$

It's important to note that when using this rule, the order of terms under multiplication does not affect the result due to the commutative property of multiplication. Thus, any permutation of the factors yields the same mathematical value. Hence, the expression can also be written as:

$8^5 \times 6^5 \times 9^5$
$8^5 \times 9^5 \times 6^5$
$9^5 \times 8^5 \times 6^5$

Upon reviewing the answer choices, each choice represents a valid permutation of $9^5 \times 6^5 \times 8^5$ .

This leads us to conclude that all given options are correct, including the explicit choice stating this fact.

Therefore, the solution to the problem is: All answers are correct.

Final Answer

All answers are correct.

Key Points to Remember

Essential concepts to master this topic

Power Rule: $(a \times b \times c)^n = a^n \times b^n \times c^n$
Technique: Apply exponent to each factor: $(9 \times 6 \times 8)^5 = 9^5 \times 6^5 \times 8^5$
Check: Any order of factors gives same result due to commutative property ✓

Common Mistakes

Avoid these frequent errors

Applying exponent to only one factor
Don't write $(9 \times 6 \times 8)^5 = 9^5 \times 6 \times 8$ = wrong answer! The exponent must apply to every single factor inside the parentheses. Always distribute the exponent to each factor: $9^5 \times 6^5 \times 8^5$ .

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 I rearrange the factors in any order?

Because multiplication is commutative! This means $a \times b = b \times a$ . So $9^5 \times 6^5 \times 8^5$ equals $8^5 \times 6^5 \times 9^5$ and any other arrangement.

Do I need to calculate the actual numbers?

Not necessarily! The question asks for the expression, not the numerical value. Leaving it as $9^5 \times 6^5 \times 8^5$ is perfectly correct and often preferred.

What if there were more factors inside the parentheses?

The same rule applies! For $(a \times b \times c \times d)^n$ , you get $a^n \times b^n \times c^n \times d^n$ . Every factor gets raised to the power.

Can I simplify before applying the exponent?

You could calculate $9 \times 6 \times 8 = 432$ first to get $432^5$ , but that's usually harder to work with than separate factors.

How do I remember this power rule?

Think of it as distributing the exponent! Just like you distribute multiplication over addition, you distribute exponents over multiplication inside parentheses.

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