Calculate (2×11)^5: Evaluating Powers of Products

Q: Why can't I just calculate 22^5 directly?

You absolutely can! $ (2\times11)^5 = 22^5 $ gives the same numerical answer. But the question asks for the expression form using the power of a product rule, which is $ 2^5 \times 11^5 $.

Q: What's the difference between (2×11)^5 and 2×11^5?

The parentheses make a huge difference! $ (2\times11)^5 $ means the entire product gets raised to the 5th power, while $ 2\times11^5 $ means only 11 gets raised to the 5th power, then multiplied by 2.

Q: How do I remember the power of a product rule?

Think of it as distributing the exponent! Just like distributing multiplication: $ a(b+c) = ab + ac $, we distribute exponents: $ (ab)^n = a^n b^n $. The exponent goes to each factor inside.

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

Yes! For example, $ (2\times3\times5)^4 = 2^4 \times 3^4 \times 5^4 $. The exponent distributes to every single factor inside the parentheses.

Q: What if I get confused about which answer choice is correct?

Calculate both! Compute $ 2^5 \times 11^5 = 32 \times 161051 $ and compare with other options. The power of a product rule should give you the same result as $ 22^5 $.

Exponent Rules with Product Expressions

Choose the expression that corresponds to the following:

$\left(2\times11\right)^5=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:08 Let's simplify the expression together.

00:14 To open parentheses with an exponent, especially when there's multiplication involved,

00:20 we raise each factor inside to the power separately.

00:24 We'll apply this formula to solve our problem.

00:32 And here is how we find the solution. Great job!

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:

$\left(2\times11\right)^5=$

Step-by-step solution

To solve the expression, we can apply the rule for the power of a product, which states that $\left(a \times b\right)^n = a^n \times b^n$ .

In this case, our expression is $\left(2\times11\right)^5$ , where $a = 2$ and $b = 11$ , and $n = 5$ .

Applying the power of a product rule gives us:

$a^n = 2^5$
$b^n = 11^5$

Therefore, $\left(2\times11\right)^5 = 2^5 \times 11^5$ .

Final Answer

$2^5\times11^5$

Key Points to Remember

Essential concepts to master this topic

Rule: $(a \times b)^n = a^n \times b^n$ - distribute exponents to each factor
Technique: Apply exponent 5 separately: $(2\times11)^5 = 2^5 \times 11^5$
Check: Verify $2^5 \times 11^5 = 32 \times 161051 = 5153632$ and $22^5 = 5153632$ ✓

Common Mistakes

Avoid these frequent errors

Squaring the product instead of distributing the exponent
Don't calculate $22^5 \times 22^5 = 22^{10}$ ! This multiplies the base by itself an extra time, giving a much larger result than needed. Always distribute the exponent to each factor inside the parentheses: $(2\times11)^5 = 2^5 \times 11^5$ .

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 calculate 22^5 directly?

You absolutely can! $(2\times11)^5 = 22^5$ gives the same numerical answer. But the question asks for the expression form using the power of a product rule, which is $2^5 \times 11^5$ .

What's the difference between (2×11)^5 and 2×11^5?

The parentheses make a huge difference! $(2\times11)^5$ means the entire product gets raised to the 5th power, while $2\times11^5$ means only 11 gets raised to the 5th power, then multiplied by 2.

How do I remember the power of a product rule?

Think of it as distributing the exponent! Just like distributing multiplication: $a(b+c) = ab + ac$ , we distribute exponents: $(ab)^n = a^n b^n$ . The exponent goes to each factor inside.

Does this rule work with more than two factors?

Yes! For example, $(2\times3\times5)^4 = 2^4 \times 3^4 \times 5^4$ . The exponent distributes to every single factor inside the parentheses.

What if I get confused about which answer choice is correct?

Calculate both! Compute $2^5 \times 11^5 = 32 \times 161051$ and compare with other options. The power of a product rule should give you the same result as $22^5$ .

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