Solve (7×5×2)^(y+4): Complete the Exponential Expression

Q: Why do I need to apply the exponent to all three numbers?

The power of a product rule requires you to raise each factor to the given power. When you have $(7 \times 5 \times 2)^{y+4}$, the exponent affects the entire product, so every number gets that exponent!

Q: What's the difference between $(7 \times 5 \times 2)^{y+4}$ and $7 \times 5 \times 2^{y+4}$ ?

In the first expression, parentheses group everything before applying the exponent. In the second, only the 2 is raised to the power $y+4$. The parentheses make a huge difference!

Q: Can I simplify $7 \times 5 \times 2$ first before applying the exponent?

You could calculate $7 \times 5 \times 2 = 70$ to get $70^{y+4}$, but the question asks for the expanded form with each factor separated.

Q: How do I remember which factors get the exponent?

Look for what's inside the parentheses! Everything grouped together by parentheses gets raised to the power. Think: "What's in the group? Everything in the group gets the exponent!"

Power of Products with Variable Exponents

Insert the corresponding expression:

$\left(7\times5\times2\right)^{y+4}=$

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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'll raise each factor to the power

00:12 We'll apply this formula to our exercise

00:15 Note that our exponent is actually the sum and it's the entire power (N)

00:20 Therefore we'll raise each factor to this power

00:30 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:

$\left(7\times5\times2\right)^{y+4}=$

Step-by-step solution

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

Step 1: Identify the current expression and its structure.
Step 2: Apply the power of a product rule.
Step 3: Simplify the expression by distributing the exponent to each factor.

Now, let's work through each step:
Step 1: The given expression is $(7 \times 5 \times 2)^{y+4}$ . This is a product inside the parentheses raised to a power.
Step 2: According to the power of a product rule, $(a \times b \times c)^n = a^n \times b^n \times c^n$ . We can apply this rule here.
Step 3: Applying the rule, the expression becomes $(7)^{y+4} \times (5)^{y+4} \times (2)^{y+4}$ .

Therefore, the correctly expanded expression is $7^{y+4}\times5^{y+4}\times2^{y+4}$ .

Final Answer

$7^{y+4}\times5^{y+4}\times2^{y+4}$

Key Points to Remember

Essential concepts to master this topic

Rule: Apply the power of a product rule: $(abc)^n = a^n \times b^n \times c^n$
Technique: Distribute the exponent $y+4$ to each factor: 7, 5, and 2
Check: Verify each factor has the same exponent $y+4$ ✓

Common Mistakes

Avoid these frequent errors

Distributing the exponent to only one factor
Don't write $7 \times 5 \times 2^{y+4}$ by applying the exponent to just one number = wrong expansion! This violates the power of a product rule. Always distribute the exponent to every single factor inside the parentheses.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do I need to apply the exponent to all three numbers?

The power of a product rule requires you to raise each factor to the given power. When you have $(7 \times 5 \times 2)^{y+4}$ , the exponent affects the entire product, so every number gets that exponent!

What's the difference between $(7 \times 5 \times 2)^{y+4}$ and $7 \times 5 \times 2^{y+4}$ ?

In the first expression, parentheses group everything before applying the exponent. In the second, only the 2 is raised to the power $y+4$ . The parentheses make a huge difference!

Can I simplify $7 \times 5 \times 2$ first before applying the exponent?

You could calculate $7 \times 5 \times 2 = 70$ to get $70^{y+4}$ , but the question asks for the expanded form with each factor separated.

How do I remember which factors get the exponent?

Look for what's inside the parentheses! Everything grouped together by parentheses gets raised to the power. Think: "What's in the group? Everything in the group gets the exponent!"

What if the exponent was just a number instead of $y+4$ ?

The rule works exactly the same! Whether the exponent is 3, $y+4$ , or $2x-1$ , you distribute it to every factor inside the parentheses.

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Solve (7×5×2)^(y+4): Complete the Exponential Expression

Power of Products with Variable Exponents

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why do I need to apply the exponent to all three numbers?

What's the difference between $(7 \times 5 \times 2)^{y+4}$ and $7 \times 5 \times 2^{y+4}$ ?

Can I simplify $7 \times 5 \times 2$ first before applying the exponent?

How do I remember which factors get the exponent?

What if the exponent was just a number instead of $y+4$ ?

🌟 Unlock Your Math Potential

More Questions

Power of a Product

Complete the Expression: (6×7×10) Raised to (y+4+a) Power

Complete the Expression: (3×6×4)^(2a) Power Operation

Expand the Expression: Solving (7×4×a)^5

Evaluate the Expression: Expanding (6×b)⁴

Solve (7×5×2)^(y+4): Complete the Exponential Expression

Power of Products with Variable Exponents

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why do I need to apply the exponent to all three numbers?

What's the difference between (7×5×2)y+4(7 \times 5 \times 2)^{y+4}(7×5×2)y+4 and 7×5×2y+47 \times 5 \times 2^{y+4}7×5×2y+4?

Can I simplify 7×5×27 \times 5 \times 27×5×2 first before applying the exponent?

How do I remember which factors get the exponent?

What if the exponent was just a number instead of y+4y+4y+4?

🌟 Unlock Your Math Potential

More Questions

Power of a Product

Complete the Expression: (6×7×10) Raised to (y+4+a) Power

Complete the Expression: (3×6×4)^(2a) Power Operation

Expand the Expression: Solving (7×4×a)^5

Evaluate the Expression: Expanding (6×b)⁴

What's the difference between $(7 \times 5 \times 2)^{y+4}$ and $7 \times 5 \times 2^{y+4}$ ?

Can I simplify $7 \times 5 \times 2$ first before applying the exponent?

What if the exponent was just a number instead of $y+4$ ?