Solve (12×5×4)^10: Evaluating a Product Raised to the Tenth Power

Q: Why can't I just multiply 12×5×4 first, then raise to the 10th power?

You absolutely can do that! $ (12 \times 5 \times 4)^{10} = (240)^{10} $. However, the question asks for the equivalent expression using the power of a product rule, not the final numerical answer.

Q: How do I remember which factors get the exponent?

Every factor inside the parentheses gets the exponent! Think of it like distributing: the exponent 10 gets distributed to 12, to 5, and to 4. No factor gets left out.

Q: What if one of the answer choices had 20 instead of 4?

That would be incorrect! You can't change the original factors. If the problem has $ 12 \times 5 \times 4 $, your answer must show $ 12^{10} \times 5^{10} \times 4^{10} $ - not any other numbers.

Q: Does order matter in the final answer?

No, order doesn't matter for multiplication! $ 12^{10} \times 5^{10} \times 4^{10} $ equals $ 5^{10} \times 4^{10} \times 12^{10} $. Just make sure all three factors have the exponent 10.

Q: Can I use this rule with different exponents?

Absolutely! The power of a product rule works with any exponent. $ (a \times b)^3 = a^3 \times b^3 $, $ (x \times y)^{50} = x^{50} \times y^{50} $, and so on.

Power of a Product with Multiple Factors

Choose the expression that corresponds to the following:

$\left(12\times5\times4\right)^{10}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:10 Let's simplify this math problem together!

00:13 When multiplying numbers with the same exponent, N,

00:18 Each number is raised to the power of N.

00:22 Let's apply this rule to our problem now.

00:32 And just like that, we've solved it!

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:

$\left(12\times5\times4\right)^{10}=$

Step-by-step solution

To solve the expression $\left(12 \times 5 \times 4\right)^{10}$ , we apply the power of a product rule of exponents, which states that when you have a product inside a power, you can apply the exponent to each factor in the product separately. This can be expressed by the formula:

$(a \times b \times c)^n = a^n \times b^n \times c^n$

In the given expression, the base is the product $12 \times 5 \times 4$ and the exponent is $10$ .

Therefore, according to the power of a product rule, the expression can be rewritten by raising each individual base to the power of $10$ :

Raise 12 to the 10th power: $12^{10}$
Raise 5 to the 10th power: $5^{10}$
Raise 4 to the 10th power: $4^{10}$

Thus, the expression $\left(12 \times 5 \times 4\right)^{10}$ simplifies to:

$12^{10} \times 5^{10} \times 4^{10}$

This shows the application of the power of a product rule for exponents by distributing the 10th power to each term within the parentheses.

Final Answer

$12^{10}\times5^{10}\times4^{10}$

Key Points to Remember

Essential concepts to master this topic

Rule: Apply exponent to each factor in the product separately
Technique: $(a \times b \times c)^n = a^n \times b^n \times c^n$
Check: Each base appears with same exponent: $12^{10} \times 5^{10} \times 4^{10}$ ✓

Common Mistakes

Avoid these frequent errors

Not applying exponent to all factors
Don't leave some factors without the exponent like $12 \times 5^{10} \times 4^{10}$ = partial application! This violates the power of a product rule and gives an incorrect expression. 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 can't I just multiply 12×5×4 first, then raise to the 10th power?

You absolutely can do that! $(12 \times 5 \times 4)^{10} = (240)^{10}$ . However, the question asks for the equivalent expression using the power of a product rule, not the final numerical answer.

How do I remember which factors get the exponent?

Every factor inside the parentheses gets the exponent! Think of it like distributing: the exponent 10 gets distributed to 12, to 5, and to 4. No factor gets left out.

What if one of the answer choices had 20 instead of 4?

That would be incorrect! You can't change the original factors. If the problem has $12 \times 5 \times 4$ , your answer must show $12^{10} \times 5^{10} \times 4^{10}$ - not any other numbers.

Does order matter in the final answer?

No, order doesn't matter for multiplication! $12^{10} \times 5^{10} \times 4^{10}$ equals $5^{10} \times 4^{10} \times 12^{10}$ . Just make sure all three factors have the exponent 10.

Can I use this rule with different exponents?

Absolutely! The power of a product rule works with any exponent. $(a \times b)^3 = a^3 \times b^3$ , $(x \times y)^{50} = x^{50} \times y^{50}$ , and so on.

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