Complete the Expression: Finding (y×a) Raised to the Power of 5

Q: Why can't I just multiply by 5 instead of raising to the 5th power?

The symbol $ ^5 $ means exponent, not multiplication! $ (y \times a)^5 $ means multiply $ (y \times a) $ by itself 5 times, which equals $ y^5 \times a^5 $.

Q: Do I apply the exponent to both variables or just one?

Both variables! The power of a product rule says every factor inside the parentheses gets raised to the same power. So $ (y \times a)^5 = y^5 \times a^5 $.

Q: What if there were three variables like (x×y×a)⁵?

Same rule applies! You'd get $ x^5 \times y^5 \times a^5 $. Every single factor inside the parentheses gets raised to the 5th power.

Q: How do I remember this rule?

Think of it as "sharing the power" - the exponent outside the parentheses gets shared equally with every factor inside. Each factor receives the full exponent!

Q: Is (y⁵×a⁵)^(1/5) the same as the original expression?

Actually, yes! Taking the 5th root (power of 1/5) of $ y^5 \times a^5 $ brings you back to $ y \times a $, but it's not the expanded form we're looking for.

Power Rules with Product Expressions

Insert the corresponding expression:

$\left(y\times a\right)^5=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:03 In order to open parentheses with a multiplication operation and an outside exponent

00:07 We raise each factor to the power

00:13 We will apply this formula to our exercise

00:20 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:

$\left(y\times a\right)^5=$

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Identify the given information
Step 2: Apply the appropriate formula
Step 3: Perform the necessary calculations

Now, let's work through each step:
Step 1: The problem provides the expression $\left(y \times a\right)^5$ and asks us to write the corresponding expanded expression.
Step 2: We'll use the exponent rule for the power of a product, which states that $(xy)^n = x^n \times y^n$ .
Step 3: Applying this rule, we raise each factor inside the parentheses to the fifth power: $y^5 \times a^5$ .

Therefore, the solution to the problem is $y^5 \times a^5$ .

Among the given choices, Choice 1: $y^5 \times a^5$ is the correct expression.

Final Answer

$y^5\times a^5$

Key Points to Remember

Essential concepts to master this topic

Power of Product Rule: Distribute the exponent to each factor inside parentheses
Technique: Apply $(xy)^n = x^n \times y^n$ to get $y^5 \times a^5$
Check: Verify each factor has the same exponent as the original power ✓

Common Mistakes

Avoid these frequent errors

Adding the exponent instead of applying it to each factor
Don't write $y \times a \times 5$ by adding the exponent = completely wrong operation! This treats 5 as multiplication, not an exponent. Always raise each factor inside parentheses to the given power.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why can't I just multiply by 5 instead of raising to the 5th power?

The symbol $^5$ means exponent, not multiplication! $(y \times a)^5$ means multiply $(y \times a)$ by itself 5 times, which equals $y^5 \times a^5$ .

Do I apply the exponent to both variables or just one?

Both variables! The power of a product rule says every factor inside the parentheses gets raised to the same power. So $(y \times a)^5 = y^5 \times a^5$ .

What if there were three variables like (x×y×a)⁵?

Same rule applies! You'd get $x^5 \times y^5 \times a^5$ . Every single factor inside the parentheses gets raised to the 5th power.

How do I remember this rule?

Think of it as "sharing the power" - the exponent outside the parentheses gets shared equally with every factor inside. Each factor receives the full exponent!

Is (y⁵×a⁵)^(1/5) the same as the original expression?

Actually, yes! Taking the 5th root (power of 1/5) of $y^5 \times a^5$ brings you back to $y \times a$ , but it's not the expanded form we're looking for.

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