Complete the Expression: (7×8) Raised to Power (b+a)

Q: Why can't I split the exponent b+a between the two bases?

The power of a product rule requires the same exponent for each base. Splitting $ b+a $ into separate parts like $ 7^b \times 8^a $ violates this rule and gives an incorrect result.

Q: What's the difference between (7×8)^(b+a) and 7×8^(b+a)?

The parentheses make a huge difference! $ (7\times8)^{b+a} $ means the entire product is raised to the power, while $ 7\times8^{b+a} $ means only the 8 is raised to the power.

Q: Do I multiply 7×8 first, then apply the exponent?

Both approaches work! You can either: Calculate $ 7\times8=56 $, then get $ 56^{b+a} $Or expand to $ 7^{b+a}\times8^{b+a} $ using the power rule

Q: How do I remember the power of a product rule?

Think of it as "sharing the exponent" - when a product is raised to a power, each factor in the product gets that same power. Like sharing equally among friends!

Q: What if the exponent was just a single variable like 'n' instead of 'b+a'?

The rule works exactly the same way! $ (7\times8)^n = 7^n\times8^n $. The complexity of the exponent doesn't change how you apply the power of a product rule.

Power of Products with Combined Exponents

Insert the corresponding expression:

$\left(7\times8\right)^{b+a}=$

❤️ 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 In order to open parentheses with a multiplication operation and an outside exponent

00:07 We'll raise each factor to the power (N)

00:11 We apply this formula to our exercise

00:14 Note that the power (N) contains an addition operation

00:18 Therefore we'll raise each factor to this power (N)

00:22 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:

$\left(7\times8\right)^{b+a}=$

Step-by-step solution

To solve this problem, we'll follow the steps below:

First, recognize that the expression given is $(7 \times 8)^{b+a}$ . We aim to use the exponent rule known as the power of a product, which states that $(xy)^n = x^n \times y^n$ .

Applying this rule to the problem at hand, we have:

$(7 \times 8)^{b+a} = 7^{b+a} \times 8^{b+a}$ .

Thus, the expression is expanded to show each base (7 and 8) raised to the combined power of $b+a$ .

Checking the choices provided, the expanded expression matches option 2.

Therefore, the correct expression is $7^{b+a}\times8^{b+a}$ .

Final Answer

$7^{b+a}\times8^{b+a}$

Key Points to Remember

Essential concepts to master this topic

Rule: Use $(xy)^n = x^n \times y^n$ for products raised to powers
Technique: Apply exponent $b+a$ to both 7 and 8 separately
Check: Verify each base has the same exponent $b+a$ ✓

Common Mistakes

Avoid these frequent errors

Distributing exponents incorrectly to individual variables
Don't split $(7\times8)^{b+a}$ into $7^b\times8^a$ = wrong distribution! This incorrectly assigns different parts of the exponent to different bases. Always apply the entire exponent $b+a$ to each base in the product.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why can't I split the exponent b+a between the two bases?

The power of a product rule requires the same exponent for each base. Splitting $b+a$ into separate parts like $7^b \times 8^a$ violates this rule and gives an incorrect result.

What's the difference between (7×8)^(b+a) and 7×8^(b+a)?

The parentheses make a huge difference! $(7\times8)^{b+a}$ means the entire product is raised to the power, while $7\times8^{b+a}$ means only the 8 is raised to the power.

Do I multiply 7×8 first, then apply the exponent?

Both approaches work! You can either:

Calculate $7\times8=56$ , then get $56^{b+a}$
Or expand to $7^{b+a}\times8^{b+a}$ using the power rule

How do I remember the power of a product rule?

Think of it as "sharing the exponent" - when a product is raised to a power, each factor in the product gets that same power. Like sharing equally among friends!

What if the exponent was just a single variable like 'n' instead of 'b+a'?

The rule works exactly the same way! $(7\times8)^n = 7^n\times8^n$ . The complexity of the exponent doesn't change how you apply the power of a product rule.

🌟 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