Solve (9²)⁴: Evaluating Nested Power Expressions

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

The power of power rule says $ (a^m)^n = a^{m \times n} $. You're raising 9² to the 4th power, which means multiplying 9² by itself 4 times: 9² × 9² × 9² × 9². Using exponent rules, this becomes 9^(2+2+2+2) = 9^8.

Q: When do I add exponents versus multiply them?

Add exponents when multiplying same bases: $ 9^2 \times 9^4 = 9^6 $. Multiply exponents when raising a power to a power: $ (9^2)^4 = 9^8 $.

Q: What if I calculated (9²)⁴ by finding 9² first?

That works too! $ 9^2 = 81 $, so $ (9^2)^4 = 81^4 $. But this creates huge numbers. Using the power rule $ (9^2)^4 = 9^8 $ keeps the expression simpler.

Q: How do I remember which rule to use?

Look for parentheses! If you see $ (a^m)^n $ with parentheses, multiply the exponents. If you see $ a^m \times a^n $ without parentheses, add the exponents.

Q: Is 9⁸ the final answer or should I calculate it?

For this problem, $ 9^8 $ is the perfect final answer! The question asks for the corresponding expression, not the numerical value. Calculating 9⁸ = 43,046,721 isn't necessary here.

Power of Power Rule with Nested Exponents

Insert the corresponding expression:

$\left(9^2\right)^4=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simply

00:02 We'll use the power of a power formula

00:04 Any number (A) to the power of (M) to the power of (N)

00:07 Equals the same number (A) to the power of the product of exponents (M*N)

00:10 We'll use this formula in our exercise

00:13 And we'll equate the numbers with the variables in the formula

00:30 We'll keep the base and multiply the exponents

00:41 Let's calculate the product

00:44 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\left(9^2\right)^4=$

Step-by-step solution

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

Step 1: Identify the provided expression: $(9^2)^4$ .
Step 2: Apply the power of a power rule for exponents.
Step 3: Simplify by multiplying the exponents.

Now, let's work through each step:

Step 1: We have the expression $(9^2)^4$ .

Step 2: Using the power of a power rule ( $(a^m)^n = a^{m \cdot n}$ ), apply it to the expression:

$(9^2)^4 = 9^{2 \times 4}$

Step 3: Simplify by calculating the product of the exponents:

$2 \times 4 = 8$

Therefore, $(9^2)^4 = 9^8$ .

The correct expression corresponding to the given problem is $\boxed{9^8}$ .

Final Answer

$9^8$

Key Points to Remember

Essential concepts to master this topic

Rule: When raising a power to a power, multiply the exponents
Technique: $(9^2)^4 = 9^{2 \times 4} = 9^8$
Check: Verify both exponents are present: base 9, inner exponent 2, outer exponent 4 ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of multiplying
Don't add 2 + 4 = 6 to get 9^6! This confuses the power of power rule with the product rule. Always multiply exponents when you have (a^m)^n = a^(m×n).

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 power rule says $(a^m)^n = a^{m \times n}$ . You're raising 9² to the 4th power, which means multiplying 9² by itself 4 times: 9² × 9² × 9² × 9². Using exponent rules, this becomes 9^(2+2+2+2) = 9^8.

When do I add exponents versus multiply them?

Add exponents when multiplying same bases: $9^2 \times 9^4 = 9^6$ . Multiply exponents when raising a power to a power: $(9^2)^4 = 9^8$ .

What if I calculated (9²)⁴ by finding 9² first?

That works too! $9^2 = 81$ , so $(9^2)^4 = 81^4$ . But this creates huge numbers. Using the power rule $(9^2)^4 = 9^8$ keeps the expression simpler.

How do I remember which rule to use?

Look for parentheses! If you see $(a^m)^n$ with parentheses, multiply the exponents. If you see $a^m \times a^n$ without parentheses, add the exponents.

Is 9⁸ the final answer or should I calculate it?

For this problem, $9^8$ is the perfect final answer! The question asks for the corresponding expression, not the numerical value. Calculating 9⁸ = 43,046,721 isn't necessary here.

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