Calculate (11×15×4)^6: Solving a Complex Exponent Problem

Q: Why are there multiple correct answers for this problem?

Because you can group the numbers differently before applying the exponent! $ 11 \times 15 \times 4 $ equals $ 11 \times 60 $ (since 15×4=60) and also equals $ 44 \times 15 $ (since 11×4=44).

Q: How do I know which numbers I can group together?

You can group any of the factors! Multiplication is associative, meaning (a×b)×c = a×(b×c). Try different combinations: group the first two, last two, or any pair that makes calculation easier.

Q: Do I have to expand all the way to the final number?

No! The question asks for the expression, not the final calculated value. Keep it in exponent form like $ 11^6 \times 15^6 \times 4^6 $ - it's much cleaner than calculating 6606!

Q: What's the power of products rule again?

When you raise a product to a power, each factor gets that same power: $ (abc)^n = a^n b^n c^n $. Think of it as distributing the exponent to every factor inside the parentheses.

Q: Can I use this rule with more than three numbers?

Absolutely! The power of products rule works with any number of factors: $ (a \times b \times c \times d)^n = a^n \times b^n \times c^n \times d^n $. Just distribute that exponent to every single factor.

Power of Products with Multiple Equivalent Forms

Choose the expression that corresponds to the following:

$\left(11\times15\times4\right)^6=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:09 Let's simplify this problem together.

00:12 When multiplying factors with the same exponent 'N', you can write the product with the exponent 'N'.

00:21 Now, let's apply this rule to our example.

00:25 Let's move on and calculate the multiplication step by step.

00:36 Remember, the order doesn't matter in multiplication. So, these expressions are equal.

00:49 Let's apply the formula to our exercise again.

00:53 We'll use the exponent rule for multiplication once more.

00:57 Let's switch the order of the factors now.

01:05 Time to explore another possible solution.

01:10 Let's calculate the multiplication again.

01:16 Using the formula for exponents of multiplication. Here we go!

01:24 And that gives us the solution. Great job!

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:

$\left(11\times15\times4\right)^6=$

Step-by-step solution

The expression in question is $(11 \times 15 \times 4)^6$ .

Using the power of a product rule, we know that any numbers $a$ , $b$ , and $c$ can be written as $(a \times b \times c)^n = a^n \times b^n \times c^n$ .

Applying this, we get:

$(11 \times 15 \times 4)^6 = 11^6 \times 15^6 \times 4^6$

Verify the multiple-choice options:
- Option 1: Clearly represents the expression as $11^6 \times 15^6 \times 4^6$ , so this is correct.
- Option 2: If we combine $15 \times 4 = 60$ , the expression becomes $(11 \times 60)^6$ , which matches $11^6 \times 60^6$ , therefore correct.
- Option 3: If we combine $11 \times 4 = 44$ , the expression becomes $(44 \times 15)^6$ , which aligns with $44^6 \times 15^6$ , therefore correct.

Since all three expressions are validated as equivalent to the original expression when simplified appropriately, all answers are correct.

Final Answer

All answers are correct.

Key Points to Remember

Essential concepts to master this topic

Power Rule: $(a \times b \times c)^n = a^n \times b^n \times c^n$
Technique: Group factors first: $15 \times 4 = 60$ or $11 \times 4 = 44$
Check: All regrouped forms must equal $11^6 \times 15^6 \times 4^6$ when expanded ✓

Common Mistakes

Avoid these frequent errors

Not recognizing equivalent grouped forms
Don't assume only one answer is correct when multiple expressions represent the same product! Students often miss that 11×60 and 44×15 are just regrouped versions of 11×15×4. Always check if different groupings create equivalent expressions using the power of products rule.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why are there multiple correct answers for this problem?

Because you can group the numbers differently before applying the exponent! $11 \times 15 \times 4$ equals $11 \times 60$ (since 15×4=60) and also equals $44 \times 15$ (since 11×4=44).

How do I know which numbers I can group together?

You can group any of the factors! Multiplication is associative, meaning (a×b)×c = a×(b×c). Try different combinations: group the first two, last two, or any pair that makes calculation easier.

Do I have to expand all the way to the final number?

No! The question asks for the expression, not the final calculated value. Keep it in exponent form like $11^6 \times 15^6 \times 4^6$ - it's much cleaner than calculating 660⁶!

What's the power of products rule again?

When you raise a product to a power, each factor gets that same power: $(abc)^n = a^n b^n c^n$ . Think of it as distributing the exponent to every factor inside the parentheses.

Can I use this rule with more than three numbers?

Absolutely! The power of products rule works with any number of factors: $(a \times b \times c \times d)^n = a^n \times b^n \times c^n \times d^n$ . Just distribute that exponent to every single factor.

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