Expand the following equation:(9x11x2x7)9

$ 11^9\times2^9\times7^9\times9^9 $

$ 2^9\times11^9\times9^9\times7^9 $

Expand (9×11×2×7)^9: Product Raised to Power Expression

Q: Why can the factors be written in different orders?

The commutative property of multiplication tells us that a × b = b × a. So $9^9 \times 11^9 \times 2^9 \times 7^9$ equals $2^9 \times 7^9 \times 9^9 \times 11^9$ - they're the same!

Q: Do I need to calculate the actual numerical answer?

Not for this type of problem! The question asks you to expand the expression, which means showing it in its distributed form: $9^9 \times 11^9 \times 2^9 \times 7^9$.

Q: How do I remember the Power of Product rule?

Think of it as sharing the exponent! When factors are multiplied together and raised to a power, each factor gets its own copy of that exponent.

Q: What's the difference between this and adding exponents?

Adding exponents is for same bases: $a^m \times a^n = a^{m+n}$. Here we have different bases (9, 11, 2, 7) so we use the Power of Product rule instead!

Exponent Rules with Product Expressions

Expand the following equation:

(9x11x2x7)⁹

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:04 According to the laws of exponents, a multiplication raised to a power (N)

00:08 equals the multiplication where each factor is raised to the same power (N)

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

00:21 We will apply this formula to our exercise

00:28 We'll break down the multiplication into each factor separately raised to the power (N)

00:39 This is one possible solution to the question

00:50 In multiplication, the order of factors doesn't matter

01:00 Therefore, we'll change the order of factors in order to determine the other solutions

01:14 This is another potential solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Expand the following equation:

(9x11x2x7)⁹

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Identify the expression to be expanded: $(9 \times 11 \times 2 \times 7)^9$
Step 2: Apply the Power of a Product rule: $(a \times b \times c \times d)^n = a^n \times b^n \times c^n \times d^n$
Step 3: Distribute the power of 9 to each factor: $9^9 \times 11^9 \times 2^9 \times 7^9$
Step 4: Recognize that due to the commutative property of multiplication, the order of factors is irrelevant.

Applying the Power of a Product rule:

$(9 \times 11 \times 2 \times 7)^9 = 9^9 \times 11^9 \times 2^9 \times 7^9$

This means that all given answer choices that represent this expression and its commutative orderings are correct solutions. Therefore, the solution to the problem is correctly all answers are equivalent.

Therefore, the correct answer to this problem is: All answers are correct.

Final Answer

All answers are correct

Key Points to Remember

Essential concepts to master this topic

Power of Product Rule: $(a \times b \times c)^n = a^n \times b^n \times c^n$
Technique: Distribute exponent 9 to each factor: $9^9 \times 11^9 \times 2^9 \times 7^9$
Check: All arrangements are equal due to commutative property ✓

Common Mistakes

Avoid these frequent errors

Applying exponent to only the first or last factor
Don't raise just 9 to the power 9 and leave other factors unchanged = $9^9 \times 11 \times 2 \times 7$ ! This ignores the parentheses and breaks the power rule. Always distribute the exponent to every single factor inside the parentheses.

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 the factors be written in different orders?

The commutative property of multiplication tells us that a × b = b × a. So $9^9 \times 11^9 \times 2^9 \times 7^9$ equals $2^9 \times 7^9 \times 9^9 \times 11^9$ - they're the same!

What if I forget to distribute the exponent to all factors?

You'll get a completely wrong answer! Remember: when you see parentheses with an exponent, that exponent applies to everything inside the parentheses.

Do I need to calculate the actual numerical answer?

Not for this type of problem! The question asks you to expand the expression, which means showing it in its distributed form: $9^9 \times 11^9 \times 2^9 \times 7^9$ .

How do I remember the Power of Product rule?

Think of it as sharing the exponent! When factors are multiplied together and raised to a power, each factor gets its own copy of that exponent.

What's the difference between this and adding exponents?

Adding exponents is for same bases: $a^m \times a^n = a^{m+n}$ . Here we have different bases (9, 11, 2, 7) so we use the Power of Product rule instead!

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