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

Q: Why does each number get the same exponent?

Because of the power of a product rule! When you have $ (abc)^n $, the exponent applies to everything inside the parentheses equally. Think of it as multiplying the entire product by itself $ n $ times.

Q: What if the exponent was just a number like 3?

The same rule applies! $ (6 \times 7 \times 10)^3 = 6^3 \times 7^3 \times 10^3 $. Whether the exponent is a number, variable, or expression like $ y+4+a $, distribute it to each factor.

Q: Can I simplify the bases first, like 6×7×10 = 420?

You could calculate $ 420^{y+4+a} $, but that's usually harder to work with! Keeping the factors separate as $ 6^{y+4+a} \times 7^{y+4+a} \times 10^{y+4+a} $ is more useful for further calculations.

Q: Does this work with addition inside parentheses too?

No! This rule only works for multiplication. For example, $ (6+7)^2 \neq 6^2 + 7^2 $. Addition has different rules for exponents.

Q: How do I remember this rule?

Think: "Exponents are distributed to products, not sums." When you see multiplication inside parentheses with an exponent outside, give each factor the same exponent!

Power of Products with Variable Exponents

Insert the corresponding expression:

$\left(6\times7\times10\right)^{y+4+a}=$

❤️ 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

00:04 In order to open parentheses with a multiplication operation and an outside exponent

00:08 Raise each factor to the power

00:14 We will apply this formula to our exercise

00:17 Note that our power is actually the sum and the power (N)

00:21 Therefore we will raise each factor to this power

00:37 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:

$\left(6\times7\times10\right)^{y+4+a}=$

Step-by-step solution

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

Step 1: Identify each factor inside the parentheses: $6$ , $7$ , and $10$ .
Step 2: Use the power of a product rule, $(abc)^n = a^n \times b^n \times c^n$ , to distribute the exponent $y + 4 + a$ to each factor.
Step 3: Rewrite the expression with each base raised to the power of $y + 4 + a$ .

Now, let's work through each step:
Step 1: We have $6 \times 7 \times 10$ as the base of the expression inside the parentheses.
Step 2: According to the power of a product rule, we distribute $y + 4 + a$ across each base, resulting in $6^{y+4+a} \times 7^{y+4+a} \times 10^{y+4+a}$ .
Step 3: Therefore, the expression $(6 \times 7 \times 10)^{y+4+a}$ expands to $6^{y+4+a} \times 7^{y+4+a} \times 10^{y+4+a}$ .

Therefore, the solution to the problem is $6^{y+4+a}\times7^{y+4+a}\times10^{y+4+a}$ .

Final Answer

$6^{y+4+a}\times7^{y+4+a}\times10^{y+4+a}$

Key Points to Remember

Essential concepts to master this topic

Rule: Distribute exponents to each factor in a product
Technique: Apply $(abc)^n = a^n \times b^n \times c^n$ to each base
Check: Each base gets same exponent: $y+4+a$ appears three times ✓

Common Mistakes

Avoid these frequent errors

Applying exponent to only one factor
Don't write $6 \times 7 \times 10^{y+4+a}$ = leaves 6 and 7 without exponents! This violates the power of a product rule. Always apply the exponent $y+4+a$ 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 does each number get the same exponent?

Because of the power of a product rule! When you have $(abc)^n$ , the exponent applies to everything inside the parentheses equally. Think of it as multiplying the entire product by itself $n$ times.

What if the exponent was just a number like 3?

The same rule applies! $(6 \times 7 \times 10)^3 = 6^3 \times 7^3 \times 10^3$ . Whether the exponent is a number, variable, or expression like $y+4+a$ , distribute it to each factor.

Can I simplify the bases first, like 6×7×10 = 420?

You could calculate $420^{y+4+a}$ , but that's usually harder to work with! Keeping the factors separate as $6^{y+4+a} \times 7^{y+4+a} \times 10^{y+4+a}$ is more useful for further calculations.

Does this work with addition inside parentheses too?

No! This rule only works for multiplication. For example, $(6+7)^2 \neq 6^2 + 7^2$ . Addition has different rules for exponents.

How do I remember this rule?

Think: "Exponents are distributed to products, not sums." When you see multiplication inside parentheses with an exponent outside, give each factor the same exponent!

🌟 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