Calculate (9×10×7)^5: Evaluating Products Raised to Powers

Q: Why can't I just multiply 9×10×7 first and then raise it to the 5th power?

You could do that! $ 9\times10\times7 = 630 $, so $ 630^5 $ is correct. But the question asks for the expanded form using the power of a product rule, which gives $ 9^5\times10^5\times7^5 $.

Q: Does the order of factors matter when applying the power rule?

No! Since multiplication is commutative, $ 9^5\times10^5\times7^5 $ equals $ 7^5\times9^5\times10^5 $ or any other arrangement of these three terms.

Q: What if one of the factors was 1?

If one factor is 1, like $ (9\times1\times7)^5 $, you'd still get $ 9^5\times1^5\times7^5 $. Since $ 1^5 = 1 $, this simplifies to $ 9^5\times7^5 $.

Q: Can this rule work with negative exponents too?

Yes! The power of a product rule works for any exponent. For example, $ (9\times10\times7)^{-2} = 9^{-2}\times10^{-2}\times7^{-2} $.

Q: How do I remember which answer choice is correct?

Look for the choice where every factor has the same exponent as the original expression. In $ (9\times10\times7)^5 $, all three factors (9, 10, 7) must each be raised to the 5th power.

Power of Products with Multiple Factors

Choose the expression that corresponds to the following:

$\left(9\times10\times7\right)^5=$

❤️ 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

00:10 Let's simplify this math problem together.

00:13 When multiplying factors with the same exponent, N,

00:17 Raise each factor to the power of N.

00:23 We'll use this rule for our problem.

00:32 Here's 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:

$\left(9\times10\times7\right)^5=$

Step-by-step solution

We begin by noting that the given expression is $\left(9\times10\times7\right)^5$ . Our task is to expand this expression using the power of a product rule.

The power of a product rule states that for any real numbers $a$ , $b$ , and $c$ and a positive integer $n$ , $(a \times b \times c)^n = a^n \times b^n \times c^n$ .

Applying this rule to the given expression, we set $a = 9$ , $b = 10$ , and $c = 7$ , and $n = 5$ .

By substituting these into the power of a product formula, we have:

$a^n = 9^5$
$b^n = 10^5$
$c^n = 7^5$

Therefore, the expression $\left(9\times10\times7\right)^5$ expands to:

$9^5\times10^5\times7^5$

Final Answer

$9^5\times10^5\times7^5$

Key Points to Remember

Essential concepts to master this topic

Power Rule: When raising a product to a power, raise each factor to that power
Technique: $(9\times10\times7)^5 = 9^5\times10^5\times7^5$ by applying exponent to each
Check: Count factors: original has 3 factors, result must have 3 factors with same exponent ✓

Common Mistakes

Avoid these frequent errors

Only applying the exponent to one or some factors
Don't write $(9\times10\times7)^5 = 9^5\times10\times7$ or similar partial applications = incorrect expansion! The exponent 5 must apply to ALL factors inside the parentheses. Always raise every single factor to the given power when expanding products.

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 multiply 9×10×7 first and then raise it to the 5th power?

You could do that! $9\times10\times7 = 630$ , so $630^5$ is correct. But the question asks for the expanded form using the power of a product rule, which gives $9^5\times10^5\times7^5$ .

Does the order of factors matter when applying the power rule?

No! Since multiplication is commutative, $9^5\times10^5\times7^5$ equals $7^5\times9^5\times10^5$ or any other arrangement of these three terms.

What if one of the factors was 1?

If one factor is 1, like $(9\times1\times7)^5$ , you'd still get $9^5\times1^5\times7^5$ . Since $1^5 = 1$ , this simplifies to $9^5\times7^5$ .

Can this rule work with negative exponents too?

Yes! The power of a product rule works for any exponent. For example, $(9\times10\times7)^{-2} = 9^{-2}\times10^{-2}\times7^{-2}$ .

How do I remember which answer choice is correct?

Look for the choice where every factor has the same exponent as the original expression. In $(9\times10\times7)^5$ , all three factors (9, 10, 7) must each be raised to the 5th power.

🌟 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