Evaluate ((9×3)^-4)^-6: Nested Negative Exponents Problem

Q: Why do I multiply the exponents instead of adding them?

The power of a power rule says $ (a^m)^n = a^{m \cdot n} $. You only add exponents when multiplying terms with the same base, like $ x^2 \cdot x^3 = x^5 $.

Q: What happens when I multiply two negative numbers?

When you multiply two negative numbers, you get a positive result! So $ -4 \times -6 = +24 $. This is why our final exponent is positive.

Q: Do I need to calculate 9 × 3 first?

No! Keep it as $ (9 \times 3) $ throughout your work. The question asks for the expression form, not the numerical value. Computing $ 27^{24} $ would be extremely difficult!

Q: How can I remember the power of a power rule?

Think of it as "powers multiply" - when you see nested exponents like $ ((something)^{inner})^{outer} $, multiply inner × outer to get your final exponent.

Q: What if one of the exponents was positive?

The rule stays the same! For example, $ ((9 \times 3)^{-4})^{3} = (9 \times 3)^{-4 \times 3} = (9 \times 3)^{-12} $. Just multiply the exponents as usual.

Step-by-step video solution

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Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Insert the corresponding expression:

$\left(\left(9\times3\right)^{-4}\right)^{-6}=$

2

Step-by-step solution

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

Step 1: Identify the expression given: $\left(\left(9 \times 3\right)^{-4}\right)^{-6}$
Step 2: Apply the power of a power rule
Step 3: Simplify the expression

Now, let's work through each step:

Step 1: The given expression is $\left(\left(9 \times 3\right)^{-4}\right)^{-6}$ .

Step 2: We'll apply the power of a power rule which states that $\left(a^m\right)^n = a^{m \cdot n}$ . In our case:

The base $a$ is $9 \times 3$
The inner exponent $m$ is $-4$
The outer exponent $n$ is $-6$

By applying the formula, we get:

\left(\left(9 \times 3\right)^{-4}\right)^{-6} = \left(9 \times 3\right)^{-4 \times -6}

Step 3: Calculate the exponent:

-4 \times -6 = 24

Thus, the expression simplifies to:

(9 \times 3)^{24}

Therefore, the solution to the problem is $\left(9 \times 3\right)^{24}$ .

Examining the multiple-choice options, the correct choice is:

Choice 2: $\left(9\times3\right)^{24}$

Thus, Choice 2 is the correct answer, aligning with our calculated solution.

3

Final Answer

$\left(9\times3\right)^{24}$

Key Points to Remember

Essential concepts to master this topic

Power Rule: When raising a power to a power, multiply exponents: $(a^m)^n = a^{m \cdot n}$
Technique: Multiply inner and outer exponents: $-4 \times -6 = 24$
Check: Two negative exponents multiply to give positive result: $(9 \times 3)^{24}$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of multiplying
Don't add the exponents -4 + (-6) = -10! This gives the wrong answer $(9 \times 3)^{-10}$ . The power of a power rule requires multiplication. Always multiply exponents when raising a power to another power.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do I multiply the exponents instead of adding them?

+

The power of a power rule says $(a^m)^n = a^{m \cdot n}$ . You only add exponents when multiplying terms with the same base, like $x^2 \cdot x^3 = x^5$ .

What happens when I multiply two negative numbers?

+

When you multiply two negative numbers, you get a positive result! So $-4 \times -6 = +24$ . This is why our final exponent is positive.

Do I need to calculate 9 × 3 first?

+

No! Keep it as $(9 \times 3)$ throughout your work. The question asks for the expression form, not the numerical value. Computing $27^{24}$ would be extremely difficult!

How can I remember the power of a power rule?

+

Think of it as "powers multiply" - when you see nested exponents like $((something)^{inner})^{outer}$ , multiply inner × outer to get your final exponent.

What if one of the exponents was positive?

+

The rule stays the same! For example, $((9 \times 3)^{-4})^{3} = (9 \times 3)^{-4 \times 3} = (9 \times 3)^{-12}$ . Just multiply the exponents as usual.

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Evaluate ((9×3)^-4)^-6: Nested Negative Exponents Problem

Power Rules with Nested Negative Exponents

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