Complete the Expression: (4×6)^b Power Problem

Q: Why can't I just multiply 4 × 6 first to get 24^b?

You absolutely can! Both $ (4 \times 6)^b = 24^b $ and $ 4^b \times 6^b $ are correct. The question asks for the distributed form using the power of a product rule.

Q: When do I use the power of a product rule?

Use this rule when you have parentheses around a multiplication raised to a power. Examples: $ (3 \times 5)^2 $, $ (xy)^4 $, or $ (2 \times 7)^n $.

Q: What's the difference between (4×6)^b and 4×6^b?

Huge difference! $ (4 \times 6)^b $ means the entire product is raised to power b, while $ 4 \times 6^b $ means only the 6 is raised to power b. Parentheses matter!

Q: Can this rule work with more than two factors?

Yes! For example: $ (2 \times 3 \times 5)^a = 2^a \times 3^a \times 5^a $. The exponent distributes to every single factor inside the parentheses.

Q: How do I remember this rule?

Think: "The exponent visits each factor". When you have $ (a \times b)^n $, the exponent n must visit both a and b, giving you $ a^n \times b^n $.

Power of Product with Exponent Distribution

Insert the corresponding expression:

$\left(4\times6\right)^b=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:07 Let's simplify this problem together.

00:10 First, we need to deal with parentheses that include multiplication and an exponent outside.

00:17 To do this, raise each factor inside to that power.

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

00:25 And there you have it! That's the solution.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\left(4\times6\right)^b=$

Step-by-step solution

To solve this problem, we will follow these steps:

Identify the given expression: $(4 \times 6)^b$ .
Apply the power of a product rule: $(a \times b)^n = a^n \times b^n$ .
Separate the expression by distributing the exponent to both elements in the product.

Now, let us apply these steps:

The expression we start with is $(4 \times 6)^b$ . According to the power of a product rule, $(a \times b)^n = a^n \times b^n$ , we distribute the exponent $b$ to each base:

$(4 \times 6)^b = 4^b \times 6^b$ .

Therefore, the expression $(4 \times 6)^b$ can be rewritten as $4^b \times 6^b$ , which corresponds to choice 1.

Final Answer

$4^b\times6^b$

Key Points to Remember

Essential concepts to master this topic

Rule: Apply $(a \times b)^n = a^n \times b^n$ for product powers
Technique: Distribute exponent b to each factor: $(4 \times 6)^b = 4^b \times 6^b$
Check: Verify both factors have same exponent: $4^b \times 6^b$ not $4^b \times 6$ ✓

Common Mistakes

Avoid these frequent errors

Only applying exponent to one factor
Don't write $(4 \times 6)^b = 4^b \times 6$ or $4 \times 6^b$ ! This ignores the power of a product rule and gives incorrect results. Always distribute the exponent to every 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't I just multiply 4 × 6 first to get 24^b?

You absolutely can! Both $(4 \times 6)^b = 24^b$ and $4^b \times 6^b$ are correct. The question asks for the distributed form using the power of a product rule.

When do I use the power of a product rule?

Use this rule when you have parentheses around a multiplication raised to a power. Examples: $(3 \times 5)^2$ , $(xy)^4$ , or $(2 \times 7)^n$ .

What's the difference between (4×6)^b and 4×6^b?

Huge difference! $(4 \times 6)^b$ means the entire product is raised to power b, while $4 \times 6^b$ means only the 6 is raised to power b. Parentheses matter!

Can this rule work with more than two factors?

Yes! For example: $(2 \times 3 \times 5)^a = 2^a \times 3^a \times 5^a$ . The exponent distributes to every single factor inside the parentheses.

How do I remember this rule?

Think: "The exponent visits each factor". When you have $(a \times b)^n$ , the exponent n must visit both a and b, giving you $a^n \times b^n$ .

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