Multiply Powers: 16^23 × 17^23 × 2^23 × 8^23 × 4^23 Expression

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

You can combine them because all exponents are the same (23)! The rule $ a^n \times b^n = (a \times b)^n $ works whenever the powers match, regardless of what the bases are.

Q: What if the exponents were different numbers?

If exponents don't match, you cannot use this rule! For example, $ 16^{23} \times 17^{24} $ cannot be simplified this way. The exponents must be identical.

Q: Do I need to calculate 16 × 17 × 2 × 8 × 4 first?

No! The question asks for the expression, not the final numerical answer. Just write $ (16 \times 17 \times 2 \times 8 \times 4)^{23} $ as your final form.

Q: Can I rearrange the order of the numbers inside the parentheses?

Yes! Since multiplication is commutative, you can write the bases in any order: $ (2 \times 4 \times 8 \times 16 \times 17)^{23} $ is equally correct.

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

Adding exponents happens when you have the same base: $ 2^3 \times 2^4 = 2^7 $. Here we have different bases with same exponents, so we group the bases instead!

Exponent Rules with Same Powers

Insert the corresponding expression:

$16^{23}\times17^{23}\times2^{23}\times8^{23}\times4^{23}=$

❤️ 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 According to the laws of exponents, a product raised to a power (N)

00:08 Equals a product where each factor is raised to that same power (N)

00:12 This formula is valid regardless of how many factors are in the product

00:18 We will apply this formula to our exercise

00:24 We'll break down the product into each factor separately raised to the power (N)

00:34 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:

$16^{23}\times17^{23}\times2^{23}\times8^{23}\times4^{23}=$

Step-by-step solution

To solve this problem, follow these steps:

Step 1: Recognize the common power in the expression.
The given expression is $16^{23} \times 17^{23} \times 2^{23} \times 8^{23} \times 4^{23}$ .
Step 2: Apply the power of a product rule.
According to the exponent rule $a^n \times b^n = (a \times b)^n$ , terms raised to the same power can be combined.
Therefore, combine the terms inside a single power: $(16 \times 17 \times 2 \times 8 \times 4)^{23}$ .

The correct answer from the given choices is: Choice 2: $\left(16 \times 17 \times 2 \times 8 \times 4\right)^{23}$ .

Final Answer

$\left(16\times17\times2\times8\times4\right)^{23}$

Key Points to Remember

Essential concepts to master this topic

Power Rule: When bases have same exponent, combine as $(a \times b)^n$
Technique: $16^{23} \times 17^{23} = (16 \times 17)^{23}$ using product rule
Check: Verify all terms have exponent 23, then group bases together ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of combining bases
Don't change $16^{23} \times 17^{23}$ to $16^{46} \times 17^{46}$ ! This adds exponents when bases are different, which is completely wrong. Always use $a^n \times b^n = (a \times b)^n$ when exponents match.

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 terms when the bases are different?

You can combine them because all exponents are the same (23)! The rule $a^n \times b^n = (a \times b)^n$ works whenever the powers match, regardless of what the bases are.

What if the exponents were different numbers?

If exponents don't match, you cannot use this rule! For example, $16^{23} \times 17^{24}$ cannot be simplified this way. The exponents must be identical.

Do I need to calculate 16 × 17 × 2 × 8 × 4 first?

No! The question asks for the expression, not the final numerical answer. Just write $(16 \times 17 \times 2 \times 8 \times 4)^{23}$ as your final form.

Can I rearrange the order of the numbers inside the parentheses?

Yes! Since multiplication is commutative, you can write the bases in any order: $(2 \times 4 \times 8 \times 16 \times 17)^{23}$ is equally correct.

What's the difference between this and adding exponents?

Adding exponents happens when you have the same base: $2^3 \times 2^4 = 2^7$ . Here we have different bases with same exponents, so we group the bases 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