Calculate (2×4×6×3)^5: Product and Power Expression

Q: Why can't I just multiply 2×4×6×3 first and then raise it to the 5th power?

You could do that, but it's much harder! $ 2 \times 4 \times 6 \times 3 = 144 $, so you'd need to calculate $ 144^5 $, which is a huge number. The factored form $ 2^5 \times 4^5 \times 6^5 \times 3^5 $ is much easier to work with!

Q: What if some factors are repeated, like (2×2×3)^4?

The same rule applies! $ (2 \times 2 \times 3)^4 = 2^4 \times 2^4 \times 3^4 $. You can then combine like bases: $ 2^4 \times 2^4 = 2^8 $, so the final answer is $ 2^8 \times 3^4 $.

Q: Do I need to calculate the final numerical answer?

Not necessarily! Often in algebra, the factored form like $ 2^5 \times 4^5 \times 6^5 \times 3^5 $ is the desired answer because it shows the mathematical structure more clearly.

Q: What if the exponent is negative, like (2×4×6×3)^(-2)?

The same rule works! $ (2 \times 4 \times 6 \times 3)^{-2} = 2^{-2} \times 4^{-2} \times 6^{-2} \times 3^{-2} $. Remember that negative exponents mean reciprocals: $ a^{-n} = \frac{1}{a^n} $.

Q: How do I remember this rule?

Think of it as distributing the exponent! Just like you distribute multiplication over addition, you distribute the power over each factor in the product. Every factor gets the same exponent!

Power of Products with Multiple Factors

Insert the corresponding expression:

$\left(2\times4\times6\times3\right)^5=$

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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 raised to a power (N)

00:08 Equals a product where each factor is raised to the 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

00:28 We'll break down the product into each factor separately raised to the power (N)

00:36 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:

$\left(2\times4\times6\times3\right)^5=$

Step-by-step solution

To solve the problem $\left(2 \times 4 \times 6 \times 3\right)^5$ , we will apply the power of a product rule.

Step 1: Identify the expression inside the parenthesis:
We have $2$ , $4$ , $6$ , and $3$ as factors, so the expression is $2 \times 4 \times 6 \times 3$ .

Step 2: Apply the power of a product rule:
According to the rule, $(a \times b \times c \times d)^n = a^n \times b^n \times c^n \times d^n$ .

Using this rule, the expression becomes:
$2^5 \times 4^5 \times 6^5 \times 3^5$ .

Therefore, the expression that represents $\left(2 \times 4 \times 6 \times 3\right)^5$ is $2^5 \times 4^5 \times 6^5 \times 3^5$ .

Final Answer

$2^5\times4^5\times6^5\times3^5$

Key Points to Remember

Essential concepts to master this topic

Power Rule: Apply the exponent to each factor separately
Technique: $(a \times b \times c)^n = a^n \times b^n \times c^n$
Check: Count factors: 4 factors inside parentheses = 4 terms with exponent 5 ✓

Common Mistakes

Avoid these frequent errors

Applying the exponent to only some factors
Don't write $2^5 \times 4 \times 6 \times 3^5$ or similar partial applications = wrong answer! This violates the power rule and gives incorrect results. Always apply 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 multiply 2×4×6×3 first and then raise it to the 5th power?

You could do that, but it's much harder! $2 \times 4 \times 6 \times 3 = 144$ , so you'd need to calculate $144^5$ , which is a huge number. The factored form $2^5 \times 4^5 \times 6^5 \times 3^5$ is much easier to work with!

What if some factors are repeated, like (2×2×3)^4?

The same rule applies! $(2 \times 2 \times 3)^4 = 2^4 \times 2^4 \times 3^4$ . You can then combine like bases: $2^4 \times 2^4 = 2^8$ , so the final answer is $2^8 \times 3^4$ .

Do I need to calculate the final numerical answer?

Not necessarily! Often in algebra, the factored form like $2^5 \times 4^5 \times 6^5 \times 3^5$ is the desired answer because it shows the mathematical structure more clearly.

What if the exponent is negative, like (2×4×6×3)^(-2)?

The same rule works! $(2 \times 4 \times 6 \times 3)^{-2} = 2^{-2} \times 4^{-2} \times 6^{-2} \times 3^{-2}$ . Remember that negative exponents mean reciprocals: $a^{-n} = \frac{1}{a^n}$ .

How do I remember this rule?

Think of it as distributing the exponent! Just like you distribute multiplication over addition, you distribute the power over each factor in the product. Every factor gets the same exponent!

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