Calculate (4×3×6×5)^4: Power of a Product Expression

Q: Why does the exponent apply to each factor separately?

The Power of a Product rule says that when you raise a product to a power, you raise each factor to that power. Think of it as: $(4 \times 3 \times 6 \times 5)^4$ means multiplying the entire product by itself 4 times!

Q: Do I need to calculate the numbers inside the parentheses first?

No! The question asks for the expression, not the numerical answer. Keep it in factored form: $4^4 \times 3^4 \times 6^4 \times 5^4$ is the correct format.

Q: What if there are only 2 factors instead of 4?

The rule works the same way! $(a \times b)^n = a^n \times b^n$. Whether you have 2, 3, 4, or more factors, every single factor gets raised to the power.

Q: Does the order of the factors matter?

No! Multiplication is commutative, so $4^4 \times 3^4 \times 6^4 \times 5^4$ equals $3^4 \times 4^4 \times 5^4 \times 6^4$. The important thing is that each factor has the exponent 4.

Power of Product with Multiple Factors

Insert the corresponding expression:

$\left(4\times3\times6\times5\right)^4=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:03 According to the exponent laws, a product raised to a power (N)

00:08 equals the 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

00:29 We'll break down the product into each factor separately raised to the power (N)

00:37 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:

$\left(4\times3\times6\times5\right)^4=$

Step-by-step solution

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

Step 1: Identify the expression and factors inside the parentheses.
Step 2: Apply the "Power of a Product" rule by distributing the exponent to each factor.
Step 3: Write the expression in the expanded form.

Now, let's work through each step:
Step 1: The expression given is $(4 \times 3 \times 6 \times 5)^4$ . The factors are 4, 3, 6, and 5.
Step 2: According to the "Power of a Product" rule, we need to apply the exponent 4 to each factor individually:
$\left(4 \times 3 \times 6 \times 5\right)^4 = 4^4 \times 3^4 \times 6^4 \times 5^4$ .
Step 3: Therefore, the expression is expanded, and each base is raised to the power of 4.

The correct answer to the problem is $4^4 \times 3^4 \times 6^4 \times 5^4$ .

Final Answer

$4^4\times3^4\times6^4\times5^4$

Key Points to Remember

Essential concepts to master this topic

Rule: Power of a product distributes the exponent to each factor
Technique: $(a \times b \times c)^n = a^n \times b^n \times c^n$
Check: Each factor has the same exponent as the original parentheses ✓

Common Mistakes

Avoid these frequent errors

Missing the exponent on one or more factors
Don't write $4^4 \times 3^4 \times 6^4 \times 5$ instead of $4^4 \times 3^4 \times 6^4 \times 5^4$ ! Forgetting to apply the exponent to every single factor gives a completely wrong result. Always distribute the exponent to each and every factor inside the parentheses.

Practice Quiz

Test your knowledge with interactive questions

Choose the expression that corresponds to the following:

$$ $\left(2\times11\right)^5=$

FAQ

Everything you need to know about this question

Why does the exponent apply to each factor separately?

The Power of a Product rule says that when you raise a product to a power, you raise each factor to that power. Think of it as: $(4 \times 3 \times 6 \times 5)^4$ means multiplying the entire product by itself 4 times!

Do I need to calculate the numbers inside the parentheses first?

No! The question asks for the expression, not the numerical answer. Keep it in factored form: $4^4 \times 3^4 \times 6^4 \times 5^4$ is the correct format.

What if there are only 2 factors instead of 4?

The rule works the same way! $(a \times b)^n = a^n \times b^n$ . Whether you have 2, 3, 4, or more factors, every single factor gets raised to the power.

How can I remember to apply the exponent to every factor?

Count the factors inside the parentheses, then count the exponents in your answer. They should match! In this problem: 4 factors inside means 4 terms with exponents in the answer.

Does the order of the factors matter?

No! Multiplication is commutative, so $4^4 \times 3^4 \times 6^4 \times 5^4$ equals $3^4 \times 4^4 \times 5^4 \times 6^4$ . The important thing is that each factor has the exponent 4.

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