Question

which expression shows 2·2·2·2·2·2 written as a power? 2⁶ what is the value of 2⁶? 2⁶ = 2·2·2·2·2·2 = (2·2)·(2·2)·(2·2) = 4·4·4 = □ · 4

Explanation:

Step1: Calculate \(4 \cdot 4 \cdot 4\) step by step

First, calculate \(4 \cdot 4 = 16\).

Step2: Multiply the result by 4

Now, multiply \(16\) by \(4\), so \(16 \cdot 4 = 64\). But in the given step, we have \(4 \cdot 4 \cdot 4\), and after \(4 \cdot 4 = 16\), the next step is \(16 \cdot 4\), but in the boxed step, we need to find the value before multiplying by 4. Wait, actually, \(4 \cdot 4 \cdot 4=(4 \cdot 4)\cdot 4 = 16\cdot 4\), but in the step \(\boldsymbol{= \square \cdot 4}\), we need to find what is \(\square\) when \(4 \cdot 4 \cdot 4=\square \cdot 4\). Since \(4 \cdot 4 = 16\), so \(\square = 16\)? Wait, no, wait. Wait, \(4 \cdot 4 \cdot 4\) can be written as \((4 \cdot 4)\cdot 4\), so the first part \(4 \cdot 4 = 16\), so the box should be \(16\)? Wait, but actually, let's re - examine. The original expansion:

\(2^6=2\cdot2\cdot2\cdot2\cdot2\cdot2=(2\cdot2)\cdot(2\cdot2)\cdot(2\cdot2)=4\cdot4\cdot4\)

Now, \(4\cdot4\cdot4=(4\cdot4)\cdot4 = 16\cdot4\), so the value in the box (the \(\square\)) is \(16\)? Wait, no, wait. Wait, maybe I made a mistake. Wait, \(4\cdot4\cdot4\): first multiply \(4\) and \(4\) to get \(16\), then multiply \(16\) by \(4\). But in the step \(\boldsymbol{= \square \cdot 4}\), we have \(4\cdot4\cdot4=\square\cdot4\). So if we factor out one \(4\), we get \(\square = 4\cdot4=16\). But actually, the final value of \(2^6\) is \(64\), but in this intermediate step, the \(\square\) is \(16\)? Wait, no, wait. Wait, let's do the multiplication step by step.

First, \(2^6 = 2\times2\times2\times2\times2\times2\)

\(2\times2 = 4\), so we have three groups of \(4\) (since \(6\) factors of \(2\), grouped into \(3\) groups of \(2\) factors of \(2\) each), so \(4\times4\times4\)

Now, \(4\times4 = 16\), so \(4\times4\times4=16\times4\)

So the value in the box (the \(\square\)) is \(16\). But wait, maybe the question is to find the value of \(2^6\) by following the steps. Wait, no, the last step in the given calculation is \(=\square\cdot4\), and then we multiply by \(4\) to get the final value. But let's check the value of \(2^6\). \(2^6 = 64\). Let's check the steps:

\(2^6=2\times2\times2\times2\times2\times2 = 64\)

Grouping as \((2\times2)\times(2\times2)\times(2\times2)=4\times4\times4\)

\(4\times4 = 16\), so \(4\times4\times4=16\times4\)

Then \(16\times4 = 64\)

So in the step \(=\square\cdot4\), \(\square = 16\)

But if we are to find the value of \(2^6\), the final answer is \(64\). But the question in the image has an intermediate step with a box. Wait, maybe the user wants to find the value in the box. Let's assume that.

Step1: Analyze the expression \(4\cdot4\cdot4\)

We know that \(4\cdot4\cdot4=(4\cdot4)\cdot4\)

Step2: Calculate \(4\cdot4\)

\(4\cdot4 = 16\)

So the value in the box (the \(\square\)) is \(16\). But if we are to find the value of \(2^6\), we can continue:

Step3: Multiply \(16\) by \(4\)

\(16\times4=64\)

But since the question in the image has an intermediate step with a box, and the step is \(=\square\cdot4\) after \(=4\cdot4\cdot4\), the value of \(\square\) is \(16\). However, if we are to find the value of \(2^6\), the final answer is \(64\).

Wait, maybe the user is asking for the value of \(2^6\). Let's confirm:

\(2^6\) means 2 multiplied by itself 6 times. \(2\times2 = 4\), \(4\times2 = 8\), \(8\times2 = 16\), \(16\times2 = 32\), \(32\times2 = 64\). So \(2^6 = 64\)

Answer:

If we are to find the value of \(2^6\), the answer is \(64\). If we are to find the value in the box (the \(\square\) in \(=\square\cdot4\)), the answer is \(16\). But since the main question "What is the value of \(2^6\)?" is asked, the answer is \(64\)