Calculate the Cube Power: (25×4)³ Step-by-Step Solution

Q: Why are all three expressions equivalent?

They all equal the same value! Power of a product rule gives us $ 25^3 \times 4^3 $, commutative property lets us write it as $ 4^3 \times 25^3 $, and since $ 25 \times 4 = 100 $, we also get $ (100)^3 $.

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

When you see $ (a \times b)^n $, distribute the exponent to each factor: $ a^n \times b^n $. So $ (25 \times 4)^3 = 25^3 \times 4^3 $.

Q: Should I simplify inside the parentheses first?

Both approaches work! You can either use the power rule first, or simplify $ 25 \times 4 = 100 $ first to get $ (100)^3 $. The choice depends on what makes the calculation easier for you.

Q: Does order matter when multiplying powers?

No! Thanks to the commutative property, $ 25^3 \times 4^3 $ equals $ 4^3 \times 25^3 $. You can multiply in any order you prefer.

Power of Products with Equivalent Expressions

Choose the expression that corresponds to the following:

$\left(25\times4\right)^3=$

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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 analyse 3 possible solutions

00:08 First, we'll calculate the multiplication and then raise to the power

00:13 This is one potential solution, let's proceed to the next solution

00:17 In order to expand parentheses containing a multiplication operation with an outer exponent

00:21 Raise each factor to the power

00:25 We'll apply this formula to our exercise

00:34 This is the second potential solution, now let's review another possible solution

00:42 In multiplication, the order of factors doesn't matter

00:45 Therefore the expressions are equal

00:50 We'll apply this formula to our exercise

00:56 Now once again we'll use the formula for the multiplication of exponents

01:05 These are the three potential 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:

$\left(25\times4\right)^3=$

Step-by-step solution

To solve this problem, we will apply the "power of a product" rule, which states that if you have a product raised to a power, each factor of the product is raised to that power. We have:

$\left(25 \times 4\right)^3 = 25^3 \times 4^3$

Using the commutative property of multiplication, this can alternatively be written as:

$4^3 \times 25^3$

Additionally, observing that $25 \times 4 = 100$ , we also have:

$(100)^3$

Thus, all the given choices are equivalent expressions for the given problem based on the power of a product and basic arithmetic simplification.

$25^3 \times 4^3$
$4^3 \times 25^3$
$\left(100\right)^3$

Therefore, the correct answer is that all answers are correct.

Final Answer

All answers are correct.

Key Points to Remember

Essential concepts to master this topic

Power Rule: When raising a product to a power, raise each factor separately
Technique: $(25 \times 4)^3 = 25^3 \times 4^3$ using distributive property
Check: Verify by calculating: $25 \times 4 = 100$ , so $(100)^3$ also works ✓

Common Mistakes

Avoid these frequent errors

Thinking only one form is correct when multiple equivalent expressions exist
Don't choose just one answer when the problem asks for equivalent expressions! All forms ( $25^3 \times 4^3$ , $4^3 \times 25^3$ , and $(100)^3$ ) represent the same value due to power rules and basic arithmetic. Always recognize that mathematical expressions can have multiple equivalent forms.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why are all three expressions equivalent?

They all equal the same value! Power of a product rule gives us $25^3 \times 4^3$ , commutative property lets us write it as $4^3 \times 25^3$ , and since $25 \times 4 = 100$ , we also get $(100)^3$ .

How do I apply the power of a product rule?

When you see $(a \times b)^n$ , distribute the exponent to each factor: $a^n \times b^n$ . So $(25 \times 4)^3 = 25^3 \times 4^3$ .

Should I simplify inside the parentheses first?

Both approaches work! You can either use the power rule first, or simplify $25 \times 4 = 100$ first to get $(100)^3$ . The choice depends on what makes the calculation easier for you.

Does order matter when multiplying powers?

No! Thanks to the commutative property, $25^3 \times 4^3$ equals $4^3 \times 25^3$ . You can multiply in any order you prefer.

What if I only calculated one form - is that wrong?

Calculating one form correctly shows you understand the math! However, when a question asks for equivalent expressions or says "all answers are correct," recognizing multiple valid forms demonstrates deeper understanding.

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