Evaluate the Expression: (4×10×7)^9 Step-by-Step

Q: Why can't I just multiply 4×10×7 first and then raise to the 9th power?

You absolutely can! $ (4 \times 10 \times 7)^9 = 280^9 $ is mathematically correct. However, the question asks for the expanded form using the power of a product rule, which is $ 4^9 \times 10^9 \times 7^9 $.

Q: Do I need to raise every factor to the 9th power?

Yes, absolutely! When you have $ (a \times b \times c)^n $, the exponent n applies to each factor individually. Missing even one factor breaks the rule.

Q: What if the factors are in a different order?

Order doesn't matter in multiplication! $ 4^9 \times 10^9 \times 7^9 $ equals $ 7^9 \times 4^9 \times 10^9 $ or any other arrangement.

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

Think: "The power outside distributes to every factor inside." Just like distributing in algebra, the exponent visits each factor in the parentheses.

Power of a Product with Multiple Factors

Choose the expression that corresponds to the following:

$\left(4\times10\times7\right)^9=$

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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 (N)

00:11 We will apply this formula to our exercise

00:20 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:

$\left(4\times10\times7\right)^9=$

Step-by-step solution

To solve the problem, we need to apply the power of a product rule of exponents. This rule states that $(a \times b \times c)^n = a^n \times b^n \times c^n$ for any numbers $a$ , $b$ , and $c$ with an exponent $n$ .

Let's apply this rule to the given expression: $\left(4\times10\times7\right)^9$ :

Identify each factor inside the parentheses: 4, 10, and 7.
The exponent applied is 9.
Apply the rule: Each factor inside the parentheses is raised to the 9th power.

This gives us the expression: $4^9 \times 10^9 \times 7^9$ .

Therefore, the final expression is $4^9 \times 10^9 \times 7^9$ .

Final Answer

$4^9\times10^9\times7^9$

Key Points to Remember

Essential concepts to master this topic

Rule: $(a \times b \times c)^n = a^n \times b^n \times c^n$ for any factors
Technique: Apply exponent 9 to each factor: $4^9 \times 10^9 \times 7^9$
Check: Each original factor should have the same exponent as the original power ✓

Common Mistakes

Avoid these frequent errors

Only raising some factors to the power
Don't raise just one or two factors to the 9th power while leaving others unchanged = $4 \times 10^9 \times 7^9$ ! This violates the power of a product rule and gives an incorrect expression. Always raise every single factor inside the parentheses to the same power.

Practice Quiz

Test your knowledge with interactive questions

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

FAQ

Everything you need to know about this question

Why can't I just multiply 4×10×7 first and then raise to the 9th power?

You absolutely can! $(4 \times 10 \times 7)^9 = 280^9$ is mathematically correct. However, the question asks for the expanded form using the power of a product rule, which is $4^9 \times 10^9 \times 7^9$ .

Do I need to raise every factor to the 9th power?

Yes, absolutely! When you have $(a \times b \times c)^n$ , the exponent n applies to each factor individually. Missing even one factor breaks the rule.

What if the factors are in a different order?

Order doesn't matter in multiplication! $4^9 \times 10^9 \times 7^9$ equals $7^9 \times 4^9 \times 10^9$ or any other arrangement.

How do I remember the power of a product rule?

Think: "The power outside distributes to every factor inside." Just like distributing in algebra, the exponent visits each factor in the parentheses.

Is this the same as the distributive property?

Similar concept! The distributive property spreads addition/subtraction, while the power of a product rule spreads exponents. Both involve distributing something to multiple terms.

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