Expand (a×b)³: Converting Product to Cube Expression

Q: Why can't I just apply the exponent to one factor?

The Power of a Product rule requires the exponent to be distributed to every factor. Think of it as: $ (a \times b)^3 = (a \times b) \times (a \times b) \times (a \times b) $, which gives you three a's and three b's!

Q: What's the difference between (ab)³ and a³b³?

They're exactly the same! $ (a \times b)^3 = a^3 \times b^3 $ is the Power of Product rule in action. The parentheses version shows the original expression, while the expanded version shows the result.

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

Yes! For example: $ (a \times b \times c)^3 = a^3 \times b^3 \times c^3 $. The exponent applies to every single factor inside the parentheses.

Q: What if the exponent is different, like (ab)²?

The same rule applies! $ (a \times b)^2 = a^2 \times b^2 $. Whatever the exponent is, it gets distributed to each factor inside the parentheses.

Q: Can I use this rule backwards?

Absolutely! If you see $ x^4 \times y^4 $, you can write it as $ (x \times y)^4 $. This is called factoring out the common exponent.

Exponent Rules with Product Powers

Insert the corresponding expression:

$\left(a\times b\right)^3=$

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:03 In order to open parentheses with a multiplication operation and an outside exponent

00:06 We raise each factor to the power

00:11 We will apply this formula to our exercise

00:16 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\left(a\times b\right)^3=$

Step-by-step solution

To solve the problem $\left(a \times b\right)^3$ , we'll apply the rules of exponents, specifically the Power of a Product rule.

Step 1: Understand the Power of a Product Rule
The Power of a Product rule states that when you raise a product to an exponent, you can apply the exponent to each factor inside the parentheses individually. Mathematically, this is expressed as: $\left(a \times b\right)^n = a^n \times b^n$
Step 2: Apply the Rule to the Given Expression
Given the expression $\left(a \times b\right)^3$ , we can apply the Power of a Product rule by raising each factor inside the parentheses to the power of 3: $\left(a \times b\right)^3 = a^3 \times b^3$
Step 3: Simplify the Expression
After applying the exponent to both $a$ and $b$ , the expression simplifies to: $a^3 \times b^3$

Therefore, the corresponding expression for $\left(a \times b\right)^3$ is $a^3 \times b^3$ .

Final Answer

$a^3\times b^3$

Key Points to Remember

Essential concepts to master this topic

Power of Product Rule: Apply exponent to each factor separately
Technique: $(a \times b)^3 = a^3 \times b^3$ distributes the exponent
Check: Both factors must have same exponent as original power ✓

Common Mistakes

Avoid these frequent errors

Applying exponent to only one factor
Don't write $(a \times b)^3 = a^3 \times b$ = wrong distribution! This ignores the power rule and gives incorrect results. Always apply the exponent to every factor inside the parentheses.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why can't I just apply the exponent to one factor?

The Power of a Product rule requires the exponent to be distributed to every factor. Think of it as: $(a \times b)^3 = (a \times b) \times (a \times b) \times (a \times b)$ , which gives you three a's and three b's!

What's the difference between (ab)³ and a³b³?

They're exactly the same! $(a \times b)^3 = a^3 \times b^3$ is the Power of Product rule in action. The parentheses version shows the original expression, while the expanded version shows the result.

Does this rule work with more than two factors?

Yes! For example: $(a \times b \times c)^3 = a^3 \times b^3 \times c^3$ . The exponent applies to every single factor inside the parentheses.

What if the exponent is different, like (ab)²?

The same rule applies! $(a \times b)^2 = a^2 \times b^2$ . Whatever the exponent is, it gets distributed to each factor inside the parentheses.

Can I use this rule backwards?

Absolutely! If you see $x^4 \times y^4$ , you can write it as $(x \times y)^4$ . This is called factoring out the common exponent.

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Exponents Rules questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime