Simplify the Expression: x⁷ × 9⁷ × y⁷ Using Exponent Properties

Q: Why can I factor out the exponent 7?

Because all three terms have the same exponent! The power of product rule works backwards: if $ (abc)^n = a^n \times b^n \times c^n $, then $ a^n \times b^n \times c^n = (abc)^n $.

Q: What if the exponents were different?

If the exponents were different (like $ x^5 \times 9^7 \times y^3 $), you cannot factor them out. The power of product rule only works when all exponents are identical.

Q: Do I need to simplify 9 to 3²?

No! The question asks for the form $ (x \times 9 \times y)^7 $. Keep 9 as is unless specifically asked to factor it further. The focus is on applying the power of product rule.

Q: How do I remember this rule?

Think of it as "common exponents come outside". When you see the same exponent on multiple terms being multiplied, you can factor it out and put everything inside parentheses.

Q: Can I write the answer as (9xy)⁷?

Yes! $ (x \times 9 \times y)^7 $ and $ (9xy)^7 $ are equivalent. Both show the same three factors raised to the 7th power. The order doesn't matter in multiplication!

Power of Product with Multiple Variables

Insert the corresponding expression:

$x^7\times9^7\times y^7=$

❤️ 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 A product where each factor is raised to the power of that factor (N)

00:08 Can be converted to parentheses of the entire product raised to the power of the factor (N)

00:16 Apply this formula to 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

Insert the corresponding expression:

$x^7\times9^7\times y^7=$

Step-by-step solution

To solve the problem, we will use the Power of a Product rule as described:

Step 1: Recognize that each factor $x$ , $9$ , and $y$ is raised to the power of 7 in the expression $x^7 \times 9^7 \times y^7$ .
Step 2: Apply the exponent rule which states $(a \times b \times c)^n = a^n \times b^n \times c^n$ .
Step 3: Rewrite the given expression as a single term raised to the 7th power: $(x \times 9 \times y)^7$ .

This step successfully combines all the individual terms under one exponent using the rule. Therefore, the expression simplifies to:

$(x \times 9 \times y)^7$

Upon examining the choices, the corresponding expression matches the option $\left(x \times 9 \times y\right)^7$ .

Thus, the correct solution to the problem is $\left(x \times 9 \times y\right)^7$ .

Final Answer

$\left(x\times9\times y\right)^7$

Key Points to Remember

Essential concepts to master this topic

Rule: When all terms have same exponent, factor it out
Technique: $a^n \times b^n \times c^n = (a \times b \times c)^n$
Check: Expand $(x \times 9 \times y)^7$ back to original form ✓

Common Mistakes

Avoid these frequent errors

Adding the exponents instead of factoring them
Don't add the exponents 7 + 7 + 7 = 21 to get $(x \times 9 \times y)^{21}$ ! This confuses the power of product rule with the product of powers rule. Always factor out the common exponent when all terms have the same power.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why can I factor out the exponent 7?

Because all three terms have the same exponent! The power of product rule works backwards: if $(abc)^n = a^n \times b^n \times c^n$ , then $a^n \times b^n \times c^n = (abc)^n$ .

What if the exponents were different?

If the exponents were different (like $x^5 \times 9^7 \times y^3$ ), you cannot factor them out. The power of product rule only works when all exponents are identical.

Do I need to simplify 9 to 3²?

No! The question asks for the form $(x \times 9 \times y)^7$ . Keep 9 as is unless specifically asked to factor it further. The focus is on applying the power of product rule.

How do I remember this rule?

Think of it as "common exponents come outside". When you see the same exponent on multiple terms being multiplied, you can factor it out and put everything inside parentheses.

Can I write the answer as (9xy)⁷?

Yes! $(x \times 9 \times y)^7$ and $(9xy)^7$ are equivalent. Both show the same three factors raised to the 7th power. The order doesn't matter in multiplication!

🌟 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