Calculate (3×7×9)^8: Evaluating Product Raised to Power of 8

Q: Why does the exponent go to each number separately?

The power of a product rule states that $ (a\times b)^n = a^n \times b^n $. Think of it like distributing: the exponent 8 must be given to each factor equally!

Q: What if I just multiply 3×7×9 first, then raise to the 8th power?

That works too! $ (3\times7\times9)^8 = (189)^8 $. However, the factored form $ 3^8\times7^8\times9^8 $ is often more useful for further calculations.

Q: Does this rule work with more than 3 factors?

Absolutely! Whether you have 2 factors or 10 factors, the rule stays the same. Each factor inside the parentheses gets raised to the outside exponent.

Q: What if one of the numbers is 1?

Great question! Since $ 1^n = 1 $ for any exponent n, you'd still apply the rule but $ 1^8 = 1 $, so it doesn't change the product.

Q: Can I use this rule with variables too?

Yes! $ (x \times y \times z)^4 = x^4 \times y^4 \times z^4 $. The power of a product rule works with any factors, whether they're numbers or variables.

Power of Product with Multiple Factors

Choose the expression that corresponds to the following:

$\left(3\times7\times9\right)^8=$

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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 When we are presented with a multiplication operation where all the factors have the same exponent (N)

00:07 Each factor can be raised to the power of (N)

00:11 We will apply this formula to our exercise

00:22 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Choose the expression that corresponds to the following:

$\left(3\times7\times9\right)^8=$

Step-by-step solution

To solve the problem, we need to apply the power of a product rule of exponents, which states that when you raise a product to a power, you can distribute the exponent to each factor in the product.

Let's break it down with the given problem:

We have the expression $\left(3\times7\times9\right)^8$ . According to the power of a product rule, this expression can be rewritten by raising each individual factor inside the parentheses to the power of 8:

Take the number 3 and raise it to the power of 8: $3^8$
Take the number 7 and raise it to the power of 8: $7^8$
Take the number 9 and raise it to the power of 8: $9^8$

Now, we can use the rule to rewrite the original expression as the product of these terms:

$3^8\times7^8\times9^8$

This is the expression you obtain when you apply the power of a product rule to $\left(3\times7\times9\right)^8$ .

Final Answer

$3^8\times7^8\times9^8$

Key Points to Remember

Essential concepts to master this topic

Rule: Distribute the exponent to each factor inside parentheses
Technique: $(3\times7\times9)^8 = 3^8\times7^8\times9^8$
Check: Count factors: original has 3 factors, result has 3 factors ✓

Common Mistakes

Avoid these frequent errors

Applying exponent to only some factors
Don't raise only one or two factors to the 8th power like $3^8\times7\times9$ = wrong distribution! This violates the power rule and gives incorrect results. Always apply the exponent to every single factor inside the parentheses.

Practice Quiz

Test your knowledge with interactive questions

$$ (4^2)^3+(g^3)^4= $$

FAQ

Everything you need to know about this question

Why does the exponent go to each number separately?

The power of a product rule states that $(a\times b)^n = a^n \times b^n$ . Think of it like distributing: the exponent 8 must be given to each factor equally!

What if I just multiply 3×7×9 first, then raise to the 8th power?

That works too! $(3\times7\times9)^8 = (189)^8$ . However, the factored form $3^8\times7^8\times9^8$ is often more useful for further calculations.

Does this rule work with more than 3 factors?

Absolutely! Whether you have 2 factors or 10 factors, the rule stays the same. Each factor inside the parentheses gets raised to the outside exponent.

What if one of the numbers is 1?

Great question! Since $1^n = 1$ for any exponent n, you'd still apply the rule but $1^8 = 1$ , so it doesn't change the product.

Can I use this rule with variables too?

Yes! $(x \times y \times z)^4 = x^4 \times y^4 \times z^4$ . The power of a product rule works with any factors, whether they're numbers or variables.

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