Solve (4³)²: Double Exponent Calculation Step-by-Step

Exponent Rules with Power of Power

(43)2= (4^3)^2=

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Step-by-step written solution

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1

Understand the problem

(43)2= (4^3)^2=

2

Step-by-step solution

To solve (43)2 (4^3)^2 , we use the power of a power rule which states that (am)n=am⋅n (a^m)^n = a^{m \cdot n} .

Here, a=4 a = 4 , m=3 m = 3 , and n=2 n = 2 .

So, we calculate 43⋅2 4^{3 \cdot 2} ,

which simplifies to 46 4^6 .

3

Final Answer

46 4^6

Key Points to Remember

Essential concepts to master this topic
  • Rule: For (am)n (a^m)^n , multiply exponents to get am⋅n a^{m \cdot n}
  • Technique: (43)2=43×2=46 (4^3)^2 = 4^{3 \times 2} = 4^6 using multiplication
  • Check: Calculate 43=64 4^3 = 64 , then 642=4096 64^2 = 4096 equals 46 4^6 ✓

Common Mistakes

Avoid these frequent errors
  • Adding instead of multiplying exponents
    Don't calculate (43)2=43+2=45 (4^3)^2 = 4^{3+2} = 4^5 ! This gives 1024 instead of 4096. The addition rule only applies to am⋅an a^m \cdot a^n , not (am)n (a^m)^n . Always multiply exponents when raising a power to another power.

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FAQ

Everything you need to know about this question

Why do I multiply the exponents instead of adding them?

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The power of a power rule requires multiplication because you're repeatedly applying the base operation. Think of (43)2 (4^3)^2 as 43×43 4^3 \times 4^3 , which equals 43+3=46 4^{3+3} = 4^6 .

What's the difference between 43×42 4^3 \times 4^2 and (43)2 (4^3)^2 ?

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Great question! 43×42=43+2=45 4^3 \times 4^2 = 4^{3+2} = 4^5 (add exponents when multiplying same bases). But (43)2=43×2=46 (4^3)^2 = 4^{3 \times 2} = 4^6 (multiply exponents for power of power). Different operations, different rules!

Can I just calculate 43 4^3 first and then square it?

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Absolutely! That's actually a great way to check your work. 43=64 4^3 = 64 , then 642=4096 64^2 = 4096 . This should equal 46=4096 4^6 = 4096 ✓

Do I always leave the answer as an exponent like 46 4^6 ?

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It depends on what the question asks for! If it says "simplify" or "evaluate," calculate 46=4096 4^6 = 4096 . If it asks to "express in exponential form," then 46 4^6 is perfect.

What if the base numbers are different, like (32)4 (3^2)^4 ?

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Same rule applies! (32)4=32×4=38 (3^2)^4 = 3^{2 \times 4} = 3^8 . The power of a power rule works for any base - just multiply those exponents.

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