Multiply Powers: Calculate 11⁶ × 10⁶ × 12⁶ Step by Step

Q: Why can I combine the powers like this?

The power of a product rule states that $ a^n \times b^n = (a \times b)^n $. Since all three numbers have the same exponent (6), we can combine them into one expression!

Q: Does the order of 11, 10, and 12 matter?

No! Multiplication follows the commutative property, meaning $ 11 \times 10 \times 12 = 12 \times 11 \times 10 $. The order doesn't change the final result.

Q: What if the exponents were different?

If the exponents were different (like $ 11^5 \times 10^6 \times 12^7 $), you cannot use this rule. The power of a product rule only works when all exponents are the same.

Q: How do I know all the answer choices are correct?

Since $ (11 \times 10 \times 12)^6 $, $ (10 \times 11 \times 12)^6 $, and $ (12 \times 10 \times 11)^6 $ all represent the same mathematical expression due to commutative property, they're all equivalent!

Q: Can I calculate the actual numerical answer?

Yes! First calculate $ 11 \times 10 \times 12 = 1320 $, then find $ 1320^6 $. But for this problem, recognizing the equivalent forms is the key skill being tested.

Power Rules with Multiplication Commutivity

Choose the expression that corresponds to the following:

$11^6\times10^6\times12^6=$

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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 When we are presented with a multiplication operation where all the factors have the same exponent (N)

00:07 The entire multiplication operation can be written with exponent (N)

00:13 We will apply this formula

00:18 In multiplication, the order of factors doesn't matter, therefore the expressions are equal

00:24 We will apply this formula to our exercise, and change the order of factors

00:41 This is the 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:

$11^6\times10^6\times12^6=$

Step-by-step solution

To address this problem, let's use the power of a product rule:

We start with $11^6 \times 10^6 \times 12^6$ .

By the power of a product rule, we can combine these into a single expression: $(11 \times 10 \times 12)^6$ .

This equation satisfies the choices given, as all representations like $(11 \times 10 \times 12)^6$ , $(10 \times 11 \times 12)^6$ , and $(12 \times 10 \times 11)^6$ are equivalent due to the commutative property of multiplication.

Thus, all answers provided are correct.

Final Answer

All answers are correct.

Key Points to Remember

Essential concepts to master this topic

Power Rule: When bases have same exponent, combine using $(a \times b \times c)^n$
Technique: Apply $a^n \times b^n \times c^n = (a \times b \times c)^n$ to combine powers
Check: All arrangements like $(11 \times 10 \times 12)^6$ and $(12 \times 10 \times 11)^6$ are equal ✓

Common Mistakes

Avoid these frequent errors

Thinking order of multiplication matters in the final answer
Don't assume $(11 \times 10 \times 12)^6$ is different from $(12 \times 10 \times 11)^6$ = wrong conclusion! Multiplication is commutative, so order doesn't change the result. Always remember that rearranging factors gives the same product.

Practice Quiz

Test your knowledge with interactive questions

$$ (4^2)^3+(g^3)^4= $$

FAQ

Everything you need to know about this question

Why can I combine the powers like this?

The power of a product rule states that $a^n \times b^n = (a \times b)^n$ . Since all three numbers have the same exponent (6), we can combine them into one expression!

Does the order of 11, 10, and 12 matter?

No! Multiplication follows the commutative property, meaning $11 \times 10 \times 12 = 12 \times 11 \times 10$ . The order doesn't change the final result.

What if the exponents were different?

If the exponents were different (like $11^5 \times 10^6 \times 12^7$ ), you cannot use this rule. The power of a product rule only works when all exponents are the same.

How do I know all the answer choices are correct?

Since $(11 \times 10 \times 12)^6$ , $(10 \times 11 \times 12)^6$ , and $(12 \times 10 \times 11)^6$ all represent the same mathematical expression due to commutative property, they're all equivalent!

Can I calculate the actual numerical answer?

Yes! First calculate $11 \times 10 \times 12 = 1320$ , then find $1320^6$ . But for this problem, recognizing the equivalent forms is the key skill being tested.

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