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

$112^0=\text{?}$

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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