Calculate (9×7)⁴: Fourth Power of Product Exercise

Q: What's the difference between (a×b)ⁿ and aⁿ×bⁿ?

They're exactly the same! The power of a product rule states that $ (a \times b)^n = a^n \times b^n $. This rule lets you distribute exponents across multiplication.

Q: Does this work with more than two factors?

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

Q: What if the exponent was on the outside of just one factor?

That would be different! $ 9 \times 7^4 $ means only 7 gets raised to the 4th power, while 9 stays as is. Always check where the parentheses are placed.

Q: How do I remember this rule?

Think: "The exponent is generous - it shares with everyone inside the parentheses!" Every factor gets the same exponent when you distribute it.

Power of Products with Exponent Distribution

Choose the expression that represents the following:

$\left(9\times7\right)^4=$

❤️ 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 expression

00:03 In order to open parentheses with an exponent over multiplication

00:06 Raise each factor to the power

00:13 We will use this formula in our exercise

00:17 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Choose the expression that represents the following:

$\left(9\times7\right)^4=$

Step-by-step solution

To solve the problem, we need to apply the power of a product rule for exponents, which states that if you have a product raised to an exponent, you can apply the exponent to each factor in the product individually.

The general form of this rule is:

$(a \times b)^n = a^n \times b^n$

According to this formula, when we have the expression $(9 \times 7)^4$ we apply the exponent 4 to each factor within the parentheses.

This process results in:

Raising 9 to the power of 4: $9^4$
Raising 7 to the power of 4: $7^4$

Therefore, the expression simplifies to:

$9^4 \times 7^4$

Final Answer

$9^4\times7^4$

Key Points to Remember

Essential concepts to master this topic

Rule: Power of a product distributes to each factor separately
Technique: $(9 \times 7)^4 = 9^4 \times 7^4$ by applying exponent to each
Check: Both equal 2,541,865 when calculated separately ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of distributing them
Don't write $(9 \times 7)^4 = 9 \times 7^4$ = 21,609! This applies the exponent to only one factor. Always distribute the exponent to every factor inside the parentheses.

Practice Quiz

Test your knowledge with interactive questions

Choose the expression that corresponds to the following:

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

FAQ

Everything you need to know about this question

Why can't I just calculate 9×7=63 first, then find 63⁴?

You absolutely can! Both methods give the same answer. The question asks for the equivalent expression using the power rule, which shows $9^4 \times 7^4$ without calculating the product first.

What's the difference between (a×b)ⁿ and aⁿ×bⁿ?

They're exactly the same! The power of a product rule states that $(a \times b)^n = a^n \times b^n$ . This rule lets you distribute exponents across multiplication.

Does this work with more than two factors?

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

What if the exponent was on the outside of just one factor?

That would be different! $9 \times 7^4$ means only 7 gets raised to the 4th power, while 9 stays as is. Always check where the parentheses are placed.

How do I remember this rule?

Think: "The exponent is generous - it shares with everyone inside the parentheses!" Every factor gets the same exponent when you distribute it.

🌟 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