Evaluate (20×5)^7: Solving a Complex Exponent Expression

Q: Why are all three answer choices equivalent?

The power of product rule gives us $ 20^7 \times 5^7 $. The commutative property lets us write this as $ 5^7 \times 20^7 $. And since $ 20 \times 5 = 100 $, we also get $ 100^7 $!

Q: When should I multiply inside parentheses first?

It's usually easier to multiply inside first when the numbers are simple! $ (20 \times 5)^7 = 100^7 $ is cleaner than $ 20^7 \times 5^7 $ for calculation.

Q: Does the order of multiplication matter with exponents?

No! Multiplication is commutative, so $ 20^7 \times 5^7 = 5^7 \times 20^7 $. The order doesn't change the final result.

Q: What if there were three numbers like (2×3×5)⁴?

The same rule applies! $ (2 \times 3 \times 5)^4 = 2^4 \times 3^4 \times 5^4 $. You can also simplify first: $ (2 \times 3 \times 5)^4 = 30^4 $.

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

Think: "The exponent visits every guest at the multiplication party!" Every factor inside the parentheses gets raised to that power.

Power Rules with Multiple Expressions

Choose the expression that corresponds to the following:

$\left(20\times5\right)^7=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:03 Let's investigate possible solutions to the given problem

00:06 In order to expand parentheses containing a multiplication operation with an outside exponent

00:09 We raise each factor to the power

00:12 We'll apply this formula to our exercise

00:17 This is one potential solution, now let's review another possible way

00:27 We can calculate the multiplication and then raise to the power

00:31 This is the second potential solution , let's proceed to the third solution

00:37 In multiplication, the order of factors doesn't matter

00:40 Therefore the expressions are equal

00:43 We'll apply this formula to our exercise

00:49 Once again we'll apply the formula to simplify the exponent

00:59 These are the three possible solutions

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:

$\left(20\times5\right)^7=$

Step-by-step solution

Step 1: We start with the expression $\left(20 \times 5\right)^7$ .
Step 2: We'll apply the power of a product rule, which states $(a \times b)^n = a^n \times b^n$ . This gives us: $\left(20 \times 5\right)^7 = 20^7 \times 5^7$ .
Step 3: To verify, notice that both $20^7 \times 5^7$ and $5^7 \times 20^7$ involve the same expression due to the commutative property of multiplication. Also, we can rewrite $\left(20 \times 5\right)$ as $100$ , leading to another form: $\left(20 \times 5\right)^7 = 100^7$
Thus, both $20^7 \times 5^7$ , $5^7 \times 20^7$ , and $100^7$ are equivalent expressions for $\left(20 \times 5\right)^7$ .

Therefore, the correct answer choice is (d) "All answers are correct."

Final Answer

All answers are correct.

Key Points to Remember

Essential concepts to master this topic

Power of Product Rule: $(a \times b)^n = a^n \times b^n$ applies to any product
Technique: Simplify inside parentheses first: $(20 \times 5)^7 = 100^7$
Check: All equivalent forms give same result: $20^7 \times 5^7 = 5^7 \times 20^7 = 100^7$ ✓

Common Mistakes

Avoid these frequent errors

Applying exponent to only one factor
Don't write $(20 \times 5)^7 = 20^7 \times 5$ = wrong answer! This ignores the power of product rule and leaves one factor without the exponent. Always apply the exponent to every factor: $(20 \times 5)^7 = 20^7 \times 5^7$ .

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why are all three answer choices equivalent?

The power of product rule gives us $20^7 \times 5^7$ . The commutative property lets us write this as $5^7 \times 20^7$ . And since $20 \times 5 = 100$ , we also get $100^7$ !

When should I multiply inside parentheses first?

It's usually easier to multiply inside first when the numbers are simple! $(20 \times 5)^7 = 100^7$ is cleaner than $20^7 \times 5^7$ for calculation.

Does the order of multiplication matter with exponents?

No! Multiplication is commutative, so $20^7 \times 5^7 = 5^7 \times 20^7$ . The order doesn't change the final result.

What if there were three numbers like (2×3×5)⁴?

The same rule applies! $(2 \times 3 \times 5)^4 = 2^4 \times 3^4 \times 5^4$ . You can also simplify first: $(2 \times 3 \times 5)^4 = 30^4$ .

How do I remember the power of product rule?

Think: "The exponent visits every guest at the multiplication party!" Every factor inside the parentheses gets raised to that power.

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