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The value of \((2^4)^3\) is 4096.
To understand why \((2^4)^3\) equals 4096, let's break it down step by step. First, we need to evaluate the expression inside the parentheses: \(2^4\). This means multiplying 2 by itself 4 times: \(2 \times 2 \times 2 \times 2\). When you do the multiplication, you get 16. So, \(2^4 = 16\).
Next, we need to raise this result to the power of 3, which is written as \(16^3\). This means multiplying 16 by itself 3 times: \(16 \times 16 \times 16\). Let's do this in stages to make it easier. First, calculate \(16 \times 16\), which equals 256. Then, multiply this result by 16 again: \(256 \times 16\).
To multiply 256 by 16, you can break it down further:
\[ 256 \times 16 = 256 \times (10 + 6) = (256 \times 10) + (256 \times 6) \]
\[ 256 \times 10 = 2560 \]
\[ 256 \times 6 = 1536 \]
\[ 2560 + 1536 = 4096 \]
So, \(16^3 = 4096\). Therefore, \((2^4)^3 = 4096\).
Another way to approach this is by using the laws of indices (exponents). According to the power of a power rule, \((a^m)^n = a^{m \times n}\). Here, \(a = 2\), \(m = 4\), and \(n = 3\). So, \((2^4)^3 = 2^{4 \times 3} = 2^{12}\). Calculating \(2^{12}\) directly also gives you 4096, confirming our result.
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