Solve Nested Powers: Evaluating ((4²)³)⁵

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

The power of a power rule says $(a^m)^n = a^{m \times n}$. This is because you're multiplying the base by itself m times, then doing that whole thing n times, which equals m × n total multiplications.

Q: How do I handle three nested powers like this problem?

Work from the inside out! First apply the rule to $(4^2)^3$, then apply it again to $(4^6)^5$. Or multiply all exponents at once: 2 × 3 × 5 = 30.

Q: What's the difference between this and adding exponents?

You add exponents when multiplying same bases: $4^2 \times 4^3 = 4^5$. You multiply exponents when raising a power to another power: $(4^2)^3 = 4^6$.

Q: Can I just calculate 4² first, then work with that number?

You could, but it gets messy quickly! $4^2 = 16$, then $(16^3)^5$ involves huge numbers. It's much easier to keep the base as 4 and just multiply the exponents.

Power Rules with Nested Exponents

Insert the corresponding expression:

$\left(\left(4^2\right)^3\right)^5=$

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Understand the problem

Insert the corresponding expression:

$\left(\left(4^2\right)^3\right)^5=$

Step-by-step solution

To solve this problem, we need to simplify the expression $\left(\left(4^2\right)^3\right)^5$ using the rules of exponents.

Let's break down the problem step by step:

Step 1: Apply the power of a power rule to $(4^2)^3$ , which states that $(a^m)^n = a^{m \cdot n}$ .
Thus, $(4^2)^3 = 4^{2 \cdot 3} = 4^6$ .
Step 2: Now apply the power of a power rule again to the result we obtained in Step 1: $(4^6)^5$ .
Using the rule again, this becomes $(4^6)^5 = 4^{6 \cdot 5} = 4^{30}$ .

Therefore, the simplified expression is $4^{30}$ .

Let's verify against given choices:

Choice 1: $4^{30}$ - This is correct.
Choice 2: $4^{25}$ - Incorrect, results from missing a power multiplication.
Choice 3: $4^{10}$ - Incorrect, results from only one power application.
Choice 4: $4^{16}$ - Incorrect, possibly confusing the base calculation.

Thus, the correct answer is indeed $4^{30}$ .

Final Answer

$4^{30}$

Key Points to Remember

Essential concepts to master this topic

Power Rule: When raising a power to another power, multiply exponents
Technique: Work from inside out: $(4^2)^3 = 4^{2 \times 3} = 4^6$
Check: Count total multiplications: 2×3×5 = 30, so answer is $4^{30}$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of multiplying
Don't add the exponents 2+3+5 = 10 to get $4^{10}$ ! Addition only works for same bases being multiplied together, not powers of powers. Always multiply exponents when you have nested powers like $((a^m)^n)^p = a^{m \times n \times p}$ .

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 $(a^m)^n = a^{m \times n}$ . This is because you're multiplying the base by itself m times, then doing that whole thing n times, which equals m × n total multiplications.

How do I handle three nested powers like this problem?

Work from the inside out! First apply the rule to $(4^2)^3$ , then apply it again to $(4^6)^5$ . Or multiply all exponents at once: 2 × 3 × 5 = 30.

What's the difference between this and adding exponents?

You add exponents when multiplying same bases: $4^2 \times 4^3 = 4^5$ . You multiply exponents when raising a power to another power: $(4^2)^3 = 4^6$ .

Can I just calculate 4² first, then work with that number?

You could, but it gets messy quickly! $4^2 = 16$ , then $(16^3)^5$ involves huge numbers. It's much easier to keep the base as 4 and just multiply the exponents.

How can I check if 4³⁰ is really the right answer?

Count the total number of times 4 gets multiplied by itself. In $((4^2)^3)^5$ , you have 2 × 3 × 5 = 30 total multiplications, so $4^{30}$ is correct!

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