Complete the Expression: (4×6) Raised to Power (b+2)

Q: Why can't I just leave it as (4×6)^(b+2)?

You absolutely can! However, the question asks you to expand using exponent rules. The expanded form $4^{b+2} \times 6^{b+2}$ shows you understand how to distribute exponents.

Q: What if I wrote 4^b × 6^2 instead?

That's incorrect because you're splitting the exponent (b+2). The power of products rule says the entire exponent must be applied to each factor, not divided between them.

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

Yes! For example: $(2 \times 3 \times 5)^n = 2^n \times 3^n \times 5^n$. The exponent distributes to every factor in the product.

Q: Can I simplify 4×6 to 24 first?

You could write $24^{b+2}$, but that doesn't demonstrate the power of products rule. The question wants you to show how exponents distribute to individual factors.

Q: What happens if the exponent was just a number like 3?

Same rule applies! $(4 \times 6)^3 = 4^3 \times 6^3 = 64 \times 216$. Whether the exponent is a number or variable, it distributes to each factor.

Power of Products with Variable Exponents

Insert the corresponding expression:

$\left(4\times6\right)^{b+2}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:08 Let's simplify this problem together.

00:11 To clear the parentheses using multiplication and an outside exponent,

00:16 Raise each part inside the parentheses to the power of N.

00:21 We'll use this method for our exercise.

00:25 Notice that the whole exponent N has an addition inside.

00:30 So, we'll raise each factor to the power of N.

00:34 And there you have it! That's our solution.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\left(4\times6\right)^{b+2}=$

Step-by-step solution

To solve this problem, we will apply the power of a product rule. This rule states that when we have a product inside a power, such as $(a \times b)^n$ , it can be rewritten as $a^n \times b^n$ . Here, our expression is $(4 \times 6)^{b+2}$ .

Let's apply this rule step by step:

Identify the base of the expression: $4 \times 6$ .
The exponent applied to this base is $b + 2$ .
Apply the power of a product rule: $(4 \times 6)^{b+2} = 4^{b+2} \times 6^{b+2}$ .

By distributing the exponent $b + 2$ to each factor of the product, we successfully rewrite the expression using the laws of exponents. The rewritten expression is $4^{b+2} \times 6^{b+2}$ .

Therefore, the final answer to the problem is $4^{b+2} \times 6^{b+2}$ .

Final Answer

$4^{b+2}\times6^{b+2}$

Key Points to Remember

Essential concepts to master this topic

Rule: Distribute exponents to each factor in the product
Technique: Apply $(a \times b)^n = a^n \times b^n$ to get $4^{b+2} \times 6^{b+2}$
Check: Both factors 4 and 6 must have identical exponent (b+2) ✓

Common Mistakes

Avoid these frequent errors

Distributing exponents partially to only one factor
Don't write $4^{b+2} \times 6$ or $4 \times 6^{b+2}$ = incomplete distribution! This violates the power of products rule and gives an incorrect expression. Always apply the exponent to every single factor in the 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't I just leave it as (4×6)^(b+2)?

You absolutely can! However, the question asks you to expand using exponent rules. The expanded form $4^{b+2} \times 6^{b+2}$ shows you understand how to distribute exponents.

What if I wrote 4^b × 6^2 instead?

That's incorrect because you're splitting the exponent (b+2). The power of products rule says the entire exponent must be applied to each factor, not divided between them.

Does this rule work with more than two factors?

Yes! For example: $(2 \times 3 \times 5)^n = 2^n \times 3^n \times 5^n$ . The exponent distributes to every factor in the product.

Can I simplify 4×6 to 24 first?

You could write $24^{b+2}$ , but that doesn't demonstrate the power of products rule. The question wants you to show how exponents distribute to individual factors.

What happens if the exponent was just a number like 3?

Same rule applies! $(4 \times 6)^3 = 4^3 \times 6^3 = 64 \times 216$ . Whether the exponent is a number or variable, it distributes to each factor.

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