Calculate (9×7×6×2)^6: Complex Product Exponentiation

Q: Why can't I just calculate the product first, then raise it to the 6th power?

You could do that, but it's much harder! $ 9 \times 7 \times 6 \times 2 = 756 $, so you'd need to calculate $ 756^6 $ - that's a huge number! Using the power rule keeps numbers manageable.

Q: Do I multiply the exponents when I see $ (abc)^6 $ ?

No! You don't multiply exponents here. The rule is distribute the exponent: each factor gets raised to the 6th power separately, then you multiply the results.

Q: What if some factors are the same, like $ (6 \times 6 \times 2)^3 $ ?

The rule still applies! You'd get $ 6^3 \times 6^3 \times 2^3 $. You could then simplify to $ 6^6 \times 2^3 $ using the rule $ a^m \times a^n = a^{m+n} $.

Q: How do I remember which rule to use?

Look for parentheses with multiplication inside and an exponent outside. That's your cue to use the power of a product rule: distribute that outside exponent to each factor inside!

Q: Can I use this rule with variables too?

Absolutely! $ (2xy)^4 = 2^4 \times x^4 \times y^4 = 16x^4y^4 $. The rule works the same way whether you have numbers, variables, or both.

Power of Products with Multiple Factors

Insert the corresponding expression:

$\left(9\times7\times6\times2\right)^6=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:04 According to the laws of exponents, a product raised to a power (N)

00:09 equals a product where each factor is raised to that same power (N)

00:16 This formula is valid regardless of how many factors are in the product

00:20 We will apply this formula to our exercise

00:25 We will break down the product into each factor separately raised to the power (N)

00:35 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:

$\left(9\times7\times6\times2\right)^6=$

Step-by-step solution

Let's solve the problem by using the power of a product rule:

The given expression is $(9 \times 7 \times 6 \times 2)^6$ .
According to the rule $(a \times b \times c)^n = a^n \times b^n \times c^n$ , we need to distribute the power of 6 to each element in the product inside the brackets.
Apply the rule: $(9 \times 7 \times 6 \times 2)^6 = 9^6 \times 7^6 \times 6^6 \times 2^6$ .

Therefore, the expression simplifies to $9^6 \times 7^6 \times 6^6 \times 2^6$ .

Final Answer

$9^6\times7^6\times6^6\times2^6$

Key Points to Remember

Essential concepts to master this topic

Rule: Distribute exponent to each factor in the product
Technique: $(a \times b \times c)^n = a^n \times b^n \times c^n$
Check: Count factors: 4 in product = 4 terms in final answer ✓

Common Mistakes

Avoid these frequent errors

Only applying the exponent to one factor
Don't just raise 9 to the 6th power and leave the rest = $9^6 \times 7 \times 6 \times 2$ ! This ignores the distributive property of exponents. Always apply the exponent to every single factor 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 just calculate the product first, then raise it to the 6th power?

You could do that, but it's much harder! $9 \times 7 \times 6 \times 2 = 756$ , so you'd need to calculate $756^6$ - that's a huge number! Using the power rule keeps numbers manageable.

Do I multiply the exponents when I see $(abc)^6$ ?

No! You don't multiply exponents here. The rule is distribute the exponent: each factor gets raised to the 6th power separately, then you multiply the results.

What if some factors are the same, like $(6 \times 6 \times 2)^3$ ?

The rule still applies! You'd get $6^3 \times 6^3 \times 2^3$ . You could then simplify to $6^6 \times 2^3$ using the rule $a^m \times a^n = a^{m+n}$ .

How do I remember which rule to use?

Look for parentheses with multiplication inside and an exponent outside. That's your cue to use the power of a product rule: distribute that outside exponent to each factor inside!

Can I use this rule with variables too?

Absolutely! $(2xy)^4 = 2^4 \times x^4 \times y^4 = 16x^4y^4$ . The rule works the same way whether you have numbers, variables, or both.

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Calculate (9×7×6×2)^6: Complex Product Exponentiation

Power of Products with Multiple Factors

❤️ 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 can't I just calculate the product first, then raise it to the 6th power?

Do I multiply the exponents when I see $(abc)^6$ ?

What if some factors are the same, like $(6 \times 6 \times 2)^3$ ?

How do I remember which rule to use?

Can I use this rule with variables too?

🌟 Unlock Your Math Potential

More Questions

Power of a Product

Evaluate (5×2×6×3)^7: Product and Power Expression

Expand (9×11×2×7)^9: Product Raised to Power Expression

Simplify the Expression: 5^(y+1) × 3^(y+1) Product Rule Challenge

Simplify the Expression: 5²ᵃ × 8²ᵃ × 7²ᵃ Using Laws of Exponents

Calculate (9×7×6×2)^6: Complex Product Exponentiation

Power of Products with Multiple Factors

❤️ 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 can't I just calculate the product first, then raise it to the 6th power?

Do I multiply the exponents when I see (abc)6 (abc)^6 (abc)6?

What if some factors are the same, like (6×6×2)3 (6 \times 6 \times 2)^3 (6×6×2)3?

How do I remember which rule to use?

Can I use this rule with variables too?

🌟 Unlock Your Math Potential

More Questions

Power of a Product

Evaluate (5×2×6×3)^7: Product and Power Expression

Expand (9×11×2×7)^9: Product Raised to Power Expression

Simplify the Expression: 5^(y+1) × 3^(y+1) Product Rule Challenge

Simplify the Expression: 5²ᵃ × 8²ᵃ × 7²ᵃ Using Laws of Exponents

Do I multiply the exponents when I see $(abc)^6$ ?

What if some factors are the same, like $(6 \times 6 \times 2)^3$ ?