Solve (3×2×4×6)^(-4): Negative Exponent Calculation

Negative Exponents with Product Rules

(3×2×4×6)4= (3\times2\times4\times6)^{-4}=

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Simply
00:03 We'll use the power rule for multiplication
00:06 Any multiplication to the power of the exponent (N)
00:09 Equals each factor separately to the power of the same exponent (N)
00:12 We'll use this formula in our exercise
00:14 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

(3×2×4×6)4= (3\times2\times4\times6)^{-4}=

2

Step-by-step solution

We begin by using the power rule for parentheses.

(zt)n=zntn (z\cdot t)^n=z^n\cdot t^n

That is, the power applied to a product inside parentheses is applied to each of the terms within when the parentheses are opened,

We apply the above rule to the given problem:

(3246)4=34244464 (3\cdot2\cdot4\cdot6)^{-4}=3^{-4}\cdot2^{-4}\cdot4^{-4}\cdot6^{-4}

Therefore, the correct answer is option d.

Note:

According to the formula of the power property inside parentheses mentioned above, it might seem as though it refers to only two terms of the product inside of the parentheses, but in reality, it is also valid for the power over a multiplication of many terms inside parentheses, as was seen above.

A good exercise is to demonstrate that if the previous property is valid for a power over a product of two terms inside parentheses (as formulated above), then it is also valid for a power over several terms of the product inside parentheses (for example - three terms, etc.).

3

Final Answer

34×24×44×64 3^{-4}\times2^{-4}\times4^{-4}\times6^{-4}

Key Points to Remember

Essential concepts to master this topic
  • Power Rule: Distribute negative exponent to each factor in product
  • Technique: (3246)4=34244464 (3\cdot2\cdot4\cdot6)^{-4} = 3^{-4}\cdot2^{-4}\cdot4^{-4}\cdot6^{-4}
  • Check: Each factor has same exponent as original power ✓

Common Mistakes

Avoid these frequent errors
  • Applying exponent to only one factor
    Don't apply -4 to just one number like 34×2×4×6 3^{-4}\times2\times4\times6 = wrong answer! This ignores the power rule for products. Always distribute the exponent 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 the negative exponent affect all the numbers?

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Because of the power rule for products! When you have (ab)n (a \cdot b)^n , it equals anbn a^n \cdot b^n . The exponent must be applied to every factor inside the parentheses.

What if I just multiply 3×2×4×6 first, then apply -4?

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That would give you 1444 144^{-4} , which is mathematically correct but much harder to work with! The distributed form 34244464 3^{-4}\cdot2^{-4}\cdot4^{-4}\cdot6^{-4} is the simplified answer they want.

Does this rule work with positive exponents too?

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Absolutely! The power rule works for any exponent. For example: (25)3=2353 (2 \cdot 5)^3 = 2^3 \cdot 5^3 . Negative exponents follow the exact same pattern.

How do I remember which answer choice is correct?

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Look for the option where every single factor has the same exponent as the original power. In this case, all four numbers (3, 2, 4, 6) should have the -4 exponent.

What's the difference between the wrong answer choices?

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  • Choice A: No exponents at all
  • Choice B & C: Only one factor gets the exponent
  • Choice D: All factors get the exponent (correct!)

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