Evaluate (2×6×7×5×4×3×4)⁴: Complex Expression Challenge

Q: Why does every number in the parentheses get raised to the 4th power?

The power of product rule says that when you raise a product to a power, you must raise each factor to that power. It's like distributing the exponent to every term inside the parentheses!

Q: Can I simplify this expression further?

Yes! You could combine like terms: $ 4^4 \times 4^4 = 4^8 $ using the rule $ a^m \times a^n = a^{m+n} $. But for this question, the distributed form is the correct answer.

Q: How do I remember the power of product rule?

Think of it as "sharing the power" - the exponent outside gets shared with every factor inside. Just like sharing pizza slices with everyone at the table!

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

00:22 We will apply this formula to our exercise

00:27 We'll break down the product into each factor separately raised to the power (N)

00:40 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Insert the corresponding expression:

$\left(2\times6\times7\times5\times4\times3\times4\right)^4=$

2

Step-by-step solution

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

Step 1: Identify the given expression
Step 2: Apply the power of a product rule
Step 3: Simplify and write the final expression

Now, let's work through each step:

Step 1: The given expression is $(2 \times 6 \times 7 \times 5 \times 4 \times 3 \times 4)$ .
We are tasked with raising this entire product to the 4th power.

Step 2: The power of a product rule, $(a \times b)^n = a^n \times b^n$ , allows us to distribute the power to each factor in the product.

Step 3: Distributing the exponent of 4 to each factor in the product:

Raise 2 to the 4th power: $2^4$
Raise 6 to the 4th power: $6^4$
Raise 7 to the 4th power: $7^4$
Raise 5 to the 4th power: $5^4$
Raise 4 to the 4th power: $4^4$
Raise 3 to the 4th power: $3^4$
Raise the second 4 to the 4th power (as it appears twice in the original expression): $4^4$

By applying the exponent to each factor, the expression becomes:

$2^4 \times 6^4 \times 7^4 \times 5^4 \times 4^4 \times 3^4 \times 4^4$

Therefore, the expression $(2 \times 6 \times 7 \times 5 \times 4 \times 3 \times 4)^4$ is equal to: $2^4 \times 6^4 \times 7^4 \times 5^4 \times 4^4 \times 3^4 \times 4^4$ .

3

Final Answer

$2^4\times6^4\times7^4\times5^4\times4^4\times3^4\times4^4$

Key Points to Remember

Essential concepts to master this topic

Power of Product Rule: $(a \times b)^n = a^n \times b^n$ for all factors
Distribution: Apply exponent 4 to each factor: $2^4, 6^4, 7^4, 5^4, 4^4, 3^4, 4^4$
Verification: Count original factors (7 terms) equals final terms (7 powers) ✓

Common Mistakes

Avoid these frequent errors

Applying exponent to only some factors
Don't raise just some numbers to the 4th power while leaving others unchanged = $2^4 \times 6 \times 7^4 \times 5 \times 4^4 \times 3 \times 4^4$ ! This violates the power of product rule and gives an incorrect expression. Always apply the exponent to every single factor in the product.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why does every number in the parentheses get raised to the 4th power?

+

The power of product rule says that when you raise a product to a power, you must raise each factor to that power. It's like distributing the exponent to every term inside the parentheses!

What if a number appears twice in the original expression?

+

Each appearance gets its own exponent! In this problem, 4 appears twice, so you get $4^4 \times 4^4$ in your final answer. Don't combine them yet - just apply the rule to each factor separately.

Can I simplify this expression further?

+

Yes! You could combine like terms: $4^4 \times 4^4 = 4^8$ using the rule $a^m \times a^n = a^{m+n}$ . But for this question, the distributed form is the correct answer.

How do I remember the power of product rule?

+

Think of it as "sharing the power" - the exponent outside gets shared with every factor inside. Just like sharing pizza slices with everyone at the table!

What's the difference between this and (2×6×7×5×4×3×4)⁴ = 2×6×7×5×4×3×4×2×6×7×5×4×3×4×2×6×7×5×4×3×4×2×6×7×5×4×3×4?

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Both methods are mathematically correct! The distributed form $2^4 \times 6^4 \times 7^4 \times 5^4 \times 4^4 \times 3^4 \times 4^4$ is more organized and shows the exponent rule clearly. Writing it out four times is messier but equivalent.

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Evaluate (2×6×7×5×4×3×4)⁴: Complex Expression Challenge

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