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

$$ (4^2)^3+(g^3)^4= $$

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