Calculate (8×9×10×5×6×3×7)⁵: Complex Exponent Problem

Q: Why does each factor get raised to the 5th power?

The power of a product rule states that $ (abc)^n = a^n \times b^n \times c^n $. When you raise a product to a power, every factor must be raised to that same power.

Q: What if I forget to raise all factors to the power?

Your answer will be completely wrong! For example, if you only raise some factors to the 5th power, you're computing a different expression entirely, not the original problem.

Q: How can I remember to apply the exponent to all factors?

Think of it like distributing the exponent! Just like you distribute multiplication, you distribute the exponent to every single factor inside the parentheses.

Q: Can I rearrange the factors before applying the exponent?

Yes! Multiplication is commutative, so $ (8 \times 9 \times 10 \times 5 \times 6 \times 3 \times 7)^5 $ equals $ (3 \times 5 \times 6 \times 7 \times 8 \times 9 \times 10)^5 $. The order doesn't matter!

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 laws of exponents, a product entirely raised to a power (N)

00:09 Equals a product where each factor is raised to the same power (N)

00:14 This formula is valid regardless of how many factors are in the product

00:26 We will apply this formula to our exercise

00:31 We'll break down the product into each factor separately raised to the power (N)

00:42 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:

$\left(8\times9\times10\times5\times6\times3\times7\right)^5=$

2

Step-by-step solution

To solve this problem, we need to appropriately apply the power of a product rule:

Preliminary setup: Identify the expression within the power. We have $8\times9\times10\times5\times6\times3\times7$ raised to the 5th power.
Step 1: Apply the exponent to each individual factor according to the power of a product rule:
$\left(8\times9\times10\times5\times6\times3\times7\right)^5 = 8^5 \times 9^5 \times 10^5 \times 5^5 \times 6^5 \times 3^5 \times 7^5$
- Here, each factor inside the parentheses is raised to the 5th power individually.
Step 2: Compare this expanded expression to the choices provided.

Choice 1: $8^5 \times 9^5 \times \left(10\times5\times6\times3\times7\right)$ does not raise each to the 5th power correctly.
Choice 2: $8 \times 9 \times 10^5 \times 5^5 \times 6^5 \times 3^5 \times 7^5$ incorrectly multiplies 8 and 9 without raising them to the 5th power, so it is not correct.
Choice 3: $5 \times \left(8\times9\times10\times5\times6\times3\times7\right)$ does not involve the correct exponentiation.
Choice 4: "None of the answers are correct" seems accurate since none of the expressions match what we derived.

Therefore, after verifying all the choices, the correct answer is None of the answers are correct, based on the power of a product rule.

3

Final Answer

None of the answers are correct

Key Points to Remember

Essential concepts to master this topic

Power Rule: When raising a product to a power, apply exponent to each factor
Technique: $(a \times b \times c)^n = a^n \times b^n \times c^n$
Check: Verify each factor has the same exponent as the original power ✓

Common Mistakes

Avoid these frequent errors

Applying the exponent to only some factors
Don't raise only some factors to the 5th power while leaving others unchanged = incorrect expression! This violates the power of a product rule and gives a completely different value. 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 does each factor get raised to the 5th power?

+

The power of a product rule states that $(abc)^n = a^n \times b^n \times c^n$ . When you raise a product to a power, every factor must be raised to that same power.

What if I forget to raise all factors to the power?

+

Your answer will be completely wrong! For example, if you only raise some factors to the 5th power, you're computing a different expression entirely, not the original problem.

How can I remember to apply the exponent to all factors?

+

Think of it like distributing the exponent! Just like you distribute multiplication, you distribute the exponent to every single factor inside the parentheses.

Why is none of the given answers correct?

+

Each answer choice makes a different error: some factors aren't raised to the 5th power, or extra operations are added. The correct answer should be $8^5 \times 9^5 \times 10^5 \times 5^5 \times 6^5 \times 3^5 \times 7^5$ .

Can I rearrange the factors before applying the exponent?

+

Yes! Multiplication is commutative, so $(8 \times 9 \times 10 \times 5 \times 6 \times 3 \times 7)^5$ equals $(3 \times 5 \times 6 \times 7 \times 8 \times 9 \times 10)^5$ . The order doesn't matter!

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