Solve the Exponential Product: 11^6 × 4^6

Q: When can I use this rule with different bases?

You can only use $ a^n \times b^n = (a \times b)^n $ when the exponents are exactly the same. If exponents differ, you must calculate each term separately first.

Q: What if I have more than two terms to multiply?

The rule extends to multiple terms! For example: $ 2^4 \times 3^4 \times 5^4 = (2 \times 3 \times 5)^4 = 30^4 $. Just make sure all exponents are identical.

Q: Should I calculate the final answer?

Usually no! The question asks for the equivalent expression, not the numerical value. $ (11 \times 4)^6 $ is the correct form, even though $ 44^6 $ would be huge!

Q: Why doesn't this work when exponents are different?

Because $ a^m \times b^n $ cannot be simplified further when m ≠ n. For example, $ 2^3 \times 3^2 = 8 \times 9 = 72 $, but there's no single base-exponent form for this.

Q: Is this the same as the power of a product rule?

Yes! It's the reverse application of $ (ab)^n = a^n b^n $. We're going backwards from $ a^n b^n $ to $ (ab)^n $ to simplify the expression.

Exponent Rules with Product Simplification

Choose the expression that corresponds to the following:

$11^6\times4^6=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:07 Let's simplify this mathematical expression together.

00:12 Here's a helpful rule: when we multiply terms that share the same exponent N, we can make things simpler.

00:18 We can rewrite the entire multiplication with a single exponent N, making our expression much cleaner.

00:25 Now, let's apply this rule to our problem and see how it works.

00:29 And there we have it! That's how we simplify expressions with common exponents.

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:

$11^6\times4^6=$

Step-by-step solution

To solve the expression, we need to apply the power of a product exponent rule. This rule states that the product of two numbers raised to the same power can be expressed as the product of those numbers raised to that power. Mathematically, it's represented as: $a^n \times b^n = (a \times b)^n$ .

In our given problem, we have $11^6 \times 4^6$ .

Here, the base numbers are 11 and 4, and both are raised to the 6th power.
According to the power of a product rule, we can combine these into a single expression: $(11 \times 4)^6$ .

Thus, the expression $11^6 \times 4^6$ can be rewritten as $(11 \times 4)^6$ .

Final Answer

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

Key Points to Remember

Essential concepts to master this topic

Power Rule: When bases are different but exponents same, combine bases: $a^n \times b^n = (a \times b)^n$
Technique: Identify same exponents first: $11^6 \times 4^6 = (11 \times 4)^6 = 44^6$
Check: Verify the rule works: $2^3 \times 3^3 = 8 \times 27 = 216$ and $(2 \times 3)^3 = 6^3 = 216$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of combining bases
Don't write $11^6 \times 4^6 = 11^{12}$ or $4^{12}$ ! Adding exponents only works when bases are the same (like $11^6 \times 11^4 = 11^{10}$ ). Always check if exponents are equal first, then combine the bases.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

When can I use this rule with different bases?

You can only use $a^n \times b^n = (a \times b)^n$ when the exponents are exactly the same. If exponents differ, you must calculate each term separately first.

What if I have more than two terms to multiply?

The rule extends to multiple terms! For example: $2^4 \times 3^4 \times 5^4 = (2 \times 3 \times 5)^4 = 30^4$ . Just make sure all exponents are identical.

Should I calculate the final answer?

Usually no! The question asks for the equivalent expression, not the numerical value. $(11 \times 4)^6$ is the correct form, even though $44^6$ would be huge!

Why doesn't this work when exponents are different?

Because $a^m \times b^n$ cannot be simplified further when m ≠ n. For example, $2^3 \times 3^2 = 8 \times 9 = 72$ , but there's no single base-exponent form for this.

Is this the same as the power of a product rule?

Yes! It's the reverse application of $(ab)^n = a^n b^n$ . We're going backwards from $a^n b^n$ to $(ab)^n$ to simplify the expression.

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