Calculate (8×5×2)^7: Evaluating the Seventh Power of a Product

Q: Why can't I just multiply 8×5×2 first and then raise to the 7th power?

You absolutely can! $ 8 \times 5 \times 2 = 80 $, so $ (8 \times 5 \times 2)^7 = 80^7 $. However, the question asks for the expanded form using the power of products rule.

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

Think of it as distributing the exponent! Just like distributing multiplication, the exponent visits each factor in the parentheses. Count your factors before and after to make sure none are missed.

Q: What if one of the numbers doesn't have an exponent in the answer choices?

That's a red flag - it means that answer choice is wrong! The power of products rule requires every factor to get the exponent. No exceptions!

Q: Does the order of the factors matter in my final answer?

No, the order doesn't matter because of the commutative property. $ 8^7 \times 5^7 \times 2^7 $ equals $ 2^7 \times 8^7 \times 5^7 $ - they're the same!

Q: Can I use this rule with more than 3 factors?

Absolutely! The power of products rule works with any number of factors. Whether you have 2, 3, 5, or 10 factors, just apply the exponent to each one.

Power of Products with Multiple Factors

Choose the expression that corresponds to the following:

$\left(8\times5\times2\right)^7=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:10 Let's simplify this problem together.

00:13 When multiplying numbers, each with exponent N, we can simplify.

00:18 We do this by raising each factor to the power of N.

00:22 Let's apply this idea to solve our exercise step by step.

00:31 And here is our solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Choose the expression that corresponds to the following:

$\left(8\times5\times2\right)^7=$

Step-by-step solution

The problem involves applying the power of a product rule in exponents. This rule states that when you raise a product to an exponent, you can apply the exponent to each factor in the product separately. Mathematically, this rule is expressed as: $(a \times b \times c)^n = a^n \times b^n \times c^n$ .

We need to apply the power of a product rule to our expression.

First, identify each individual factor in the product:

Factor 1: $8$
Factor 2: $5$
Factor 3: $2$

Now, apply the exponent $7$ to each factor:

$8^7$
$5^7$
$2^7$

Therefore, the expression $(8 \times 5 \times 2)^7$ simplifies to:

$8^7 \times 5^7 \times 2^7$

Final Answer

$8^7\times5^7\times2^7$

Key Points to Remember

Essential concepts to master this topic

Rule: Apply the exponent to each factor: $(a \times b \times c)^n = a^n \times b^n \times c^n$
Technique: Distribute 7 to each factor: $(8 \times 5 \times 2)^7 = 8^7 \times 5^7 \times 2^7$
Check: Count factors - if original has 3 factors, answer should have 3 terms with exponents ✓

Common Mistakes

Avoid these frequent errors

Applying the exponent to only some factors
Don't apply 7 to just two factors like $8^7 \times 5^7 \times 2$ = wrong answer! This violates the power rule and gives an incorrect result. 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 can't I just multiply 8×5×2 first and then raise to the 7th power?

You absolutely can! $8 \times 5 \times 2 = 80$ , so $(8 \times 5 \times 2)^7 = 80^7$ . However, the question asks for the expanded form using the power of products rule.

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

Think of it as distributing the exponent! Just like distributing multiplication, the exponent visits each factor in the parentheses. Count your factors before and after to make sure none are missed.

What if one of the numbers doesn't have an exponent in the answer choices?

That's a red flag - it means that answer choice is wrong! The power of products rule requires every factor to get the exponent. No exceptions!

Does the order of the factors matter in my final answer?

No, the order doesn't matter because of the commutative property. $8^7 \times 5^7 \times 2^7$ equals $2^7 \times 8^7 \times 5^7$ - they're the same!

Can I use this rule with more than 3 factors?

Absolutely! The power of products rule works with any number of factors. Whether you have 2, 3, 5, or 10 factors, just apply the exponent to each one.

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