Simplify 2⁹ × 4⁹ × 7⁹: Product of Powers Expression

Q: Why can I combine these powers when the bases are different?

You can combine them because they all have the same exponent (9)! The rule $ a^n \times b^n = (a \times b)^n $ works backwards too: when you see matching exponents, factor them out.

Q: What if the exponents were different numbers?

If exponents don't match, you cannot combine the bases! For example, $ 2^8 \times 4^9 \times 7^{10} $ stays as is because 8, 9, and 10 are all different.

Q: Do I need to calculate 2 × 4 × 7 = 56?

Not necessarily! The question asks for the expression form, so $ (2 \times 4 \times 7)^9 $ is the correct answer. You could simplify to $ 56^9 $, but it's not required here.

Q: How do I remember this exponent rule?

Think of it as "same power, combine bases". Just like $ 3 \times 3 \times 3 = 3^3 $, when you see the same exponent repeated, you can factor it out!

Q: What's the difference between this and adding exponents?

Adding exponents ($ a^m \times a^n = a^{m+n} $) only works with the same base. Here we have different bases (2, 4, 7) but the same exponent (9), so we use the product rule instead.

Exponent Rules with Product Powers

Choose the expression that corresponds to the following:

$2^9\times4^9\times7^9=$

❤️ 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 When we are presented with a multiplication operation where all the factors have the same exponent (N)

00:07 We can write the exponent (N) over the entire product

00:14 We can apply this formula to our exercise

00:22 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 corresponds to the following:

$2^9\times4^9\times7^9=$

Step-by-step solution

The given expression is $2^9 \times 4^9 \times 7^9$ .

We need to simplify this expression by using the exponent rule for the power of a product. The rule states that $(a \times b \times c)^n = a^n \times b^n \times c^n$ , which can be inverted to simplify a product of terms with the same exponent.

Thus, the expression $2^9 \times 4^9 \times 7^9$ can be rewritten as $(2 \times 4 \times 7)^9$ .

Breaking it down:

Identify the common exponent, which is $9$ .
Combine the bases under a single power:
- The bases are $2, 4,$ .
Apply the exponent rule: $2^9 \times 4^9 \times 7^9 = (2 \times 4 \times 7)^9$ .

Therefore, the corresponding expression is $\left(2\times4\times7\right)^9$ .

Final Answer

$\left(2\times4\times7\right)^9$

Key Points to Remember

Essential concepts to master this topic

Rule: When bases have same exponent, combine under single power
Technique: $2^9 \times 4^9 \times 7^9 = (2 \times 4 \times 7)^9$
Check: Verify bases multiply correctly: 2 × 4 × 7 = 56 ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of combining bases
Don't add the exponents like $2^{9+9+9} = 2^{27}$ ! This only works when bases are the same. Always check if exponents match first, then combine the different bases under one power.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why can I combine these powers when the bases are different?

You can combine them because they all have the same exponent (9)! The rule $a^n \times b^n = (a \times b)^n$ works backwards too: when you see matching exponents, factor them out.

What if the exponents were different numbers?

If exponents don't match, you cannot combine the bases! For example, $2^8 \times 4^9 \times 7^{10}$ stays as is because 8, 9, and 10 are all different.

Do I need to calculate 2 × 4 × 7 = 56?

Not necessarily! The question asks for the expression form, so $(2 \times 4 \times 7)^9$ is the correct answer. You could simplify to $56^9$ , but it's not required here.

How do I remember this exponent rule?

Think of it as "same power, combine bases". Just like $3 \times 3 \times 3 = 3^3$ , when you see the same exponent repeated, you can factor it out!

What's the difference between this and adding exponents?

Adding exponents ( $a^m \times a^n = a^{m+n}$ ) only works with the same base. Here we have different bases (2, 4, 7) but the same exponent (9), so we use the product rule instead.

🌟 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