Solve: Product of Powers 20¹¹ × 4¹¹ × 2¹¹ × 5¹¹ × 3¹¹

Q: Why can I combine all the bases together?

Because all terms have the same exponent (11)! The product of powers rule states that $ a^n \times b^n = (a \times b)^n $, and this works for any number of terms with the same exponent.

Q: What if the exponents were different?

If the exponents were different, you cannot use this rule! For example, $ 20^{11} \times 4^{10} $ cannot be simplified to $ (20 \times 4)^{something} $. The exponents must be identical.

Q: Do I need to calculate 20 × 4 × 2 × 5 × 3?

Not necessarily! The question asks for the equivalent expression, not the final numerical answer. So $ (20 \times 4 \times 2 \times 5 \times 3)^{11} $ is the correct form.

Q: Can I rearrange the bases in any order?

Yes! Multiplication is commutative, so $ (20 \times 4 \times 2 \times 5 \times 3)^{11} $ equals $ (3 \times 5 \times 2 \times 4 \times 20)^{11} $. The order doesn't matter!

Q: What's wrong with the first answer choice?

The first choice $ 11 \times 20 \times 4 \times 2 \times 5 \times 3 $ completely ignores the exponents! It treats $ 20^{11} $ as just 20, which is completely wrong.

Product of Powers with Common Exponents

Insert the corresponding expression:

$20^{11}\times4^{11}\times2^{11}\times5^{11}\times3^{11}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:16 Let's simplify this problem together.

00:19 Using power laws, when a product is raised to a power N, pause.

00:24 Each factor of the product is raised to power N. pause.

00:30 This works no matter how many factors there are. pause.

00:41 Let's apply this rule to our exercise. pause.

00:46 We'll separate each factor and raise it to power N. pause.

01:01 And that's our solution! Well done!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$20^{11}\times4^{11}\times2^{11}\times5^{11}\times3^{11}=$

Step-by-step solution

To solve this problem, we will use the rule of exponents which allows us to simplify expressions of the form $a^n \times b^n = (a \times b)^n$ .

Here's how to simplify the expression step by step:

Step 1: Start with the original expression: $20^{11} \times 4^{11} \times 2^{11} \times 5^{11} \times 3^{11}$ .
Step 2: Recognize that each term (20, 4, 2, 5, and 3) is raised to the power of 11, which allows the use of the product of powers rule. We can combine these bases into a single base raised to the common power: $(20 \times 4 \times 2 \times 5 \times 3)^{11}$

Hence, the expression simplifies to $\left(20 \times 4 \times 2 \times 5 \times 3\right)^{11}$ , which corresponds to choice 3.

Final Answer

$\left(20\times4\times2\times5\times3\right)^{11}$

Key Points to Remember

Essential concepts to master this topic

Rule: When multiplying powers with same exponent: $a^n \times b^n = (a \times b)^n$
Technique: Combine all bases first: $20 \times 4 \times 2 \times 5 \times 3 = 2400$
Check: All terms have exponent 11, so rule applies perfectly ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of combining bases
Don't add the exponents 11 + 11 + 11 + 11 + 11 = 55! This creates the wrong expression with exponent 55. The exponents stay the same (11) because we're multiplying terms with the same power. Always combine the bases and keep the common exponent.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why can I combine all the bases together?

Because all terms have the same exponent (11)! The product of powers rule states that $a^n \times b^n = (a \times b)^n$ , and this works for any number of terms with the same exponent.

What if the exponents were different?

If the exponents were different, you cannot use this rule! For example, $20^{11} \times 4^{10}$ cannot be simplified to $(20 \times 4)^{something}$ . The exponents must be identical.

Do I need to calculate 20 × 4 × 2 × 5 × 3?

Not necessarily! The question asks for the equivalent expression, not the final numerical answer. So $(20 \times 4 \times 2 \times 5 \times 3)^{11}$ is the correct form.

Can I rearrange the bases in any order?

Yes! Multiplication is commutative, so $(20 \times 4 \times 2 \times 5 \times 3)^{11}$ equals $(3 \times 5 \times 2 \times 4 \times 20)^{11}$ . The order doesn't matter!

What's wrong with the first answer choice?

The first choice $11 \times 20 \times 4 \times 2 \times 5 \times 3$ completely ignores the exponents! It treats $20^{11}$ as just 20, which is completely wrong.

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