Calculate (16×2×3)¹¹: Solving a Compound Exponential Expression

Q: Why does every factor get raised to the 11th power?

The power of a product rule states that $ (a\times b\times c)^n = a^n \times b^n \times c^n $. When you multiply numbers together and raise the whole thing to a power, each individual factor must be raised to that same power.

Q: Can I simplify the expression inside the parentheses first?

Yes, but it's not necessary! You could calculate $ 16\times2\times3 = 96 $ to get $ 96^{11} $, but the power rule gives you the distributed form directly.

Q: How do I remember which exponent goes where?

Think of it as sharing equally: the exponent outside the parentheses gets distributed to every single factor inside. No factor gets left out!

Q: Is this the same as multiplying the exponents?

No! We're distributing the exponent, not multiplying. $ (16\times2\times3)^{11} $ becomes $ 16^{11}\times2^{11}\times3^{11} $, where each base keeps the same exponent 11.

Power of Product with Multiple Factors

Choose the expression that corresponds to the following:

$\left(16\times2\times3\right)^{11}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:10 Let's simplify this problem together.

00:13 When multiplying numbers with the same exponent, N, start by noting each number is raised to that power, N.

00:20 Think of each factor being taken to the power of N.

00:24 We'll use this rule in our example step by step. Ready to see how it works?

00:31 Here's how we find 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:

$\left(16\times2\times3\right)^{11}=$

Step-by-step solution

To solve the expression, we will use the power of a product rule. According to this rule, when you have a product raised to an exponent, you can distribute the exponent to each factor in the product. Mathematically, this is expressed as:

$(a \times b \times c)^n = a^n \times b^n \times c^n$

In our expression, $a = 16$ , $b = 2$ , and $c = 3$ .

Applying the power of a product formula to our expression gives:

$(16 \times 2 \times 3)^{11} = 16^{11} \times 2^{11} \times 3^{11}$

This shows that each factor inside the parentheses is raised to the power of 11, which is consistent with the provided answer:

$16^{11}\times2^{11}\times3^{11}$

Final Answer

$16^{11}\times2^{11}\times3^{11}$

Key Points to Remember

Essential concepts to master this topic

Power Rule: Distribute the exponent to each factor in the product
Technique: $(16\times2\times3)^{11} = 16^{11}\times2^{11}\times3^{11}$
Check: Each factor inside parentheses gets the same exponent 11 ✓

Common Mistakes

Avoid these frequent errors

Only applying exponent to some factors
Don't raise only one or two factors to the power while leaving others unchanged = $16^{11}\times2\times3$ ! This violates the power of a product rule and gives a much smaller result. Always apply 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 does every factor get raised to the 11th power?

The power of a product rule states that $(a\times b\times c)^n = a^n \times b^n \times c^n$ . When you multiply numbers together and raise the whole thing to a power, each individual factor must be raised to that same power.

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

You'll get a much smaller answer! For example, if you only raise 16 to the 11th power but leave 2 and 3 unchanged, you're calculating $16^{11}\times2\times3$ instead of the correct $16^{11}\times2^{11}\times3^{11}$ .

Can I simplify the expression inside the parentheses first?

Yes, but it's not necessary! You could calculate $16\times2\times3 = 96$ to get $96^{11}$ , but the power rule gives you the distributed form directly.

How do I remember which exponent goes where?

Think of it as sharing equally: the exponent outside the parentheses gets distributed to every single factor inside. No factor gets left out!

Is this the same as multiplying the exponents?

No! We're distributing the exponent, not multiplying. $(16\times2\times3)^{11}$ becomes $16^{11}\times2^{11}\times3^{11}$ , where each base keeps the same exponent 11.

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