Calculate (3×8×6×4×2)³: Cubing a Product Expression

Q: Why can't I just cube the first number and leave the rest alone?

Because the entire product is being cubed, not just one factor! The parentheses mean everything inside gets the exponent. Think of it like: the whole group is multiplied by itself 3 times.

Q: What if I want to calculate the actual numerical answer?

You can! First apply the rule to get $ 3^3\times8^3\times6^3\times4^3\times2^3 $, then calculate each cube: $ 27\times512\times216\times64\times8 $. But the question asks for the expression form.

Q: Does this rule work for any number of factors?

Yes! Whether you have 2 factors or 20 factors, the power of a product rule applies the same way. Each factor gets the exponent distributed to it.

Q: What if the exponent was different, like squared instead of cubed?

The same rule applies! $ (3\times8\times6\times4\times2)^2 = 3^2\times8^2\times6^2\times4^2\times2^2 $. Just change all the exponents to match the original power.

Power of Product with Multiple Factors

Insert the corresponding expression:

$\left(3\times8\times6\times4\times2\right)^3=$

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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 According to the laws of exponents, a product that is raised to the power (N)

00:07 Equals a product where each factor is raised to that same power (N)

00:13 This formula is valid regardless of how many factors are in the product

00:24 We will apply this formula to our exercise

00:41 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:

$\left(3\times8\times6\times4\times2\right)^3=$

Step-by-step solution

To solve this problem, we will use the power of a product rule, which helps distribute the exponent over each factor within the parentheses.

Step 1: Identify the expression. We have $(3 \times 8 \times 6 \times 4 \times 2)^3$ .
Step 2: Apply the power of a product rule. This rule states that if we have a product raised to an exponent, we can distribute the exponent to each factor. Mathematically, $(a \times b \times c \ldots)^n = a^n \times b^n \times c^n \ldots$ .
Step 3: Apply the rule to the expression.

We distribute the exponent 3 to each of the factors inside the parentheses:

$(3 \times 8 \times 6 \times 4 \times 2)^3 = 3^3 \times 8^3 \times 6^3 \times 4^3 \times 2^3$ .

Therefore, the corresponding expression is $3^3 \times 8^3 \times 6^3 \times 4^3 \times 2^3$ .

Final Answer

$3^3\times8^3\times6^3\times4^3\times2^3$

Key Points to Remember

Essential concepts to master this topic

Power Rule: Distribute the exponent to each factor inside parentheses
Application: $(3\times8\times6\times4\times2)^3 = 3^3\times8^3\times6^3\times4^3\times2^3$
Verification: Each original factor gets the same exponent (3) applied ✓

Common Mistakes

Avoid these frequent errors

Applying the exponent to only some factors
Don't apply the exponent 3 to just the first or last factor like $3^3\times8\times6\times4\times2$ = wrong expression! This violates the power of a product rule and changes the mathematical meaning. Always distribute the exponent to every single factor inside the parentheses.

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 cube the first number and leave the rest alone?

Because the entire product is being cubed, not just one factor! The parentheses mean everything inside gets the exponent. Think of it like: the whole group is multiplied by itself 3 times.

What if I want to calculate the actual numerical answer?

You can! First apply the rule to get $3^3\times8^3\times6^3\times4^3\times2^3$ , then calculate each cube: $27\times512\times216\times64\times8$ . But the question asks for the expression form.

Does this rule work for any number of factors?

Yes! Whether you have 2 factors or 20 factors, the power of a product rule applies the same way. Each factor gets the exponent distributed to it.

Can I group some factors together before applying the exponent?

You could, but it's not necessary and might cause confusion. The cleanest approach is to distribute the exponent to each individual factor as shown in the rule.

What if the exponent was different, like squared instead of cubed?

The same rule applies! $(3\times8\times6\times4\times2)^2 = 3^2\times8^2\times6^2\times4^2\times2^2$ . Just change all the exponents to match the original power.

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