Solve: 8⁷ × 10⁷ × 16⁷ Power Multiplication Problem

Q: Why can I combine the bases when they have the same exponent?

This follows the power of a product rule working backwards! Since $ (abc)^n = a^n \times b^n \times c^n $, we can reverse it: when we see $ a^n \times b^n \times c^n $, we can write it as $ (abc)^n $.

Q: What if the exponents were different numbers?

If exponents are different, you cannot combine the bases! For example, $ 8^5 \times 10^7 \times 16^3 $ cannot be simplified using this rule. The exponents must be exactly the same.

Q: Do I need to calculate the actual value of 8 × 10 × 16?

No! The question asks for the equivalent expression, not the numerical answer. Writing $ (8 \times 10 \times 16)^7 $ is the complete answer - you don't need to compute 1,280.

Q: How is this different from adding exponents?

Adding exponents applies when you have the same base: $ a^m \times a^n = a^{m+n} $. Here we have different bases with the same exponent, so we use the product rule instead!

Q: Can this work with more than three terms?

Absolutely! If you have $ a^n \times b^n \times c^n \times d^n $, it equals $ (abcd)^n $. The rule works for any number of terms as long as they all have the same exponent.

Power of Products with Same Exponents

Choose the expression that corresponds to the following:

$8^7\times10^7\times16^7=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:04 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

00:17 We can apply this formula in our exercise

00:24 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:

$8^7\times10^7\times16^7=$

Step-by-step solution

The given expression is $8^7\times10^7\times16^7$ . We need to apply the power of a product rule for exponents. This rule states that for any numbers $a$ , $b$ , and $c$ , if they have the same exponent $n$ , then $(a\times b\times c)^n=a^n\times b^n\times c^n$ .

In this problem, we recognize that 8, 10, and 16 all have the same exponent of 7. Therefore we can apply the rule directly:

$8^7 \times 10^7 \times 16^7$
Applying the power of a product rule:
$(8 \times 10 \times 16)^7$

This simplified form matches the pattern we recognize from the power of a product rule, verifying that $(8\times10\times16)^7$ is indeed the correct transformation of the original expression $8^7\times10^7\times16^7$ .

Final Answer

$\left(8\times10\times16\right)^7$

Key Points to Remember

Essential concepts to master this topic

Rule: When bases have same exponent, combine them first
Technique: $8^7 \times 10^7 \times 16^7 = (8 \times 10 \times 16)^7$
Check: Verify pattern matches $a^n \times b^n \times c^n = (abc)^n$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of applying product rule
Don't add the exponents to get $8 \times 10 \times 16^{21}$ = wrong operation! This confuses multiplication rules with power rules. Always recognize when bases have the same exponent and use $(abc)^n = a^n \times b^n \times c^n$ .

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why can I combine the bases when they have the same exponent?

This follows the power of a product rule working backwards! Since $(abc)^n = a^n \times b^n \times c^n$ , we can reverse it: when we see $a^n \times b^n \times c^n$ , we can write it as $(abc)^n$ .

What if the exponents were different numbers?

If exponents are different, you cannot combine the bases! For example, $8^5 \times 10^7 \times 16^3$ cannot be simplified using this rule. The exponents must be exactly the same.

Do I need to calculate the actual value of 8 × 10 × 16?

No! The question asks for the equivalent expression, not the numerical answer. Writing $(8 \times 10 \times 16)^7$ is the complete answer - you don't need to compute 1,280.

How is this different from adding exponents?

Adding exponents applies when you have the same base: $a^m \times a^n = a^{m+n}$ . Here we have different bases with the same exponent, so we use the product rule instead!

Can this work with more than three terms?

Absolutely! If you have $a^n \times b^n \times c^n \times d^n$ , it equals $(abcd)^n$ . The rule works for any number of terms as long as they all have the same exponent.

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