Solve ((4×6)³)⁴: Calculating Nested Exponents Step-by-Step

Q: Why do I multiply 3 × 4 instead of adding them?

The power rule says $ (a^m)^n = a^{m \times n} $. You're taking something that's already raised to the 3rd power and raising it to the 4th power again. That means multiplying the base by itself 3 × 4 = 12 times total!

Q: What's the difference between this and $ a^3 \times a^4 $ ?

Great question! $ a^3 \times a^4 = a^{3+4} = a^7 $ (you add when multiplying same bases). But $ (a^3)^4 = a^{3 \times 4} = a^{12} $ (you multiply when raising a power to another power).

Q: Do I need to calculate 4 × 6 first?

Not for this problem! The question asks for the equivalent expression, so keep $ (4 \times 6) $ as the base and just apply the power rule to get $ (4 \times 6)^{12} $.

Q: How can I remember when to add vs multiply exponents?

Memory trick: Multiplication → Addition ($ a^3 \times a^4 = a^{3+4} $) and Power → Multiplication ($ (a^3)^4 = a^{3 \times 4} $). Look for parentheses around the first power!

Q: What if there are three nested exponents like $ ((a^2)^3)^4 $ ?

Work from inside out! First: $ (a^2)^3 = a^{2 \times 3} = a^6 $. Then: $ (a^6)^4 = a^{6 \times 4} = a^{24} $. Or multiply all at once: $ a^{2 \times 3 \times 4} = a^{24} $!

Power Rule with Nested Exponents

Insert the corresponding expression:

$\left(\right.\left(4\times6\right)^3)^4=$

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Understand the problem

Insert the corresponding expression:

$\left(\right.\left(4\times6\right)^3)^4=$

Step-by-step solution

To solve this problem, we'll use the power of a power property of exponents, which states that for any base $a$ and exponents $m$ and $n$ , $(a^m)^n = a^{m \times n}$ .

Step 1: Identify the base and exponents:
In the given expression $\left(\left(4 \times 6\right)^3\right)^4$ , the base is $(4 \times 6)$ , the inner exponent is 3, and the outer exponent is 4.
Step 2: Apply the power of a power rule:
According to the rule, $\left((4 \times 6)^3\right)^4$ simplifies to $(4 \times 6)^{3 \times 4}$ .
Step 3: Calculate the new exponent:
Multiply the exponents: $3 \times 4 = 12$ . Hence, the expression simplifies to $(4 \times 6)^{12}$ .

The expression $\left(\left(4 \times 6\right)^3\right)^4$ is equivalent to $(4 \times 6)^{3 \times 4}$ . Therefore, the correct choice is:

$\left(4\times6\right)^{3\times4}$

Therefore, the correct answer is Choice 1.

Final Answer

$\left(4\times6\right)^{3\times4}$

Key Points to Remember

Essential concepts to master this topic

Power Rule: When raising a power to another power, multiply exponents
Technique: $(a^m)^n = a^{m \times n}$ gives $(4 \times 6)^{3 \times 4}$
Check: Verify by calculating: 3 × 4 = 12 makes $(24)^{12}$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of multiplying
Don't add 3 + 4 = 7 to get $(4 \times 6)^7$ = wrong answer! Adding only works when multiplying same bases, not raising powers to powers. Always multiply exponents when you see $(a^m)^n$ .

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do I multiply 3 × 4 instead of adding them?

The power rule says $(a^m)^n = a^{m \times n}$ . You're taking something that's already raised to the 3rd power and raising it to the 4th power again. That means multiplying the base by itself 3 × 4 = 12 times total!

What's the difference between this and $a^3 \times a^4$ ?

Great question! $a^3 \times a^4 = a^{3+4} = a^7$ (you add when multiplying same bases). But $(a^3)^4 = a^{3 \times 4} = a^{12}$ (you multiply when raising a power to another power).

Do I need to calculate 4 × 6 first?

Not for this problem! The question asks for the equivalent expression, so keep $(4 \times 6)$ as the base and just apply the power rule to get $(4 \times 6)^{12}$ .

How can I remember when to add vs multiply exponents?

Memory trick: Multiplication → Addition ( $a^3 \times a^4 = a^{3+4}$ ) and Power → Multiplication ( $(a^3)^4 = a^{3 \times 4}$ ). Look for parentheses around the first power!

What if there are three nested exponents like $((a^2)^3)^4$ ?

Work from inside out! First: $(a^2)^3 = a^{2 \times 3} = a^6$ . Then: $(a^6)^4 = a^{6 \times 4} = a^{24}$ . Or multiply all at once: $a^{2 \times 3 \times 4} = a^{24}$ !

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Solve ((4×6)³)⁴: Calculating Nested Exponents Step-by-Step

Power Rule with Nested Exponents

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why do I multiply 3 × 4 instead of adding them?

What's the difference between this and $a^3 \times a^4$ ?

Do I need to calculate 4 × 6 first?

How can I remember when to add vs multiply exponents?

What if there are three nested exponents like $((a^2)^3)^4$ ?

🌟 Unlock Your Math Potential

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Solve ((4×6)³)⁴: Calculating Nested Exponents Step-by-Step

Power Rule with Nested Exponents

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why do I multiply 3 × 4 instead of adding them?

What's the difference between this and a3×a4 a^3 \times a^4 a3×a4?

Do I need to calculate 4 × 6 first?

How can I remember when to add vs multiply exponents?

What if there are three nested exponents like ((a2)3)4 ((a^2)^3)^4 ((a2)3)4?

🌟 Unlock Your Math Potential

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What's the difference between this and $a^3 \times a^4$ ?

What if there are three nested exponents like $((a^2)^3)^4$ ?