Simplify and Solve: 2^2·2^-3·2^4 / 2^3·2^-2·2^5

Question

$2^2\cdot2^{-3}\cdot2^4\text{ }_{—\text{ }}2^3\cdot2^{-2}\cdot2^5$

00:00 Determine the appropriate the sign

00:03 Let's simplify the left side

00:10 When multiplying powers with equal bases

00:13 The power of the result equals the sum of the powers

00:19 We'll apply this formula to our exercise, let's add up the powers

00:24 This is simplifying the left side

00:29 Let's simplify the right side

00:34 We'll apply the same formula again and add up the powers

00:42 Let's identify where the power is greater

00:47 This is the solution

We start by simplifying each expression using the laws of exponents.

For the first expression $2^2 \cdot 2^{-3} \cdot 2^4$ :

Apply the multiplication of powers rule: $2^2 \cdot 2^{-3} = 2^{2 + (-3)} = 2^{-1}$ .
Now, multiply by $2^4$ : $2^{-1} \cdot 2^4 = 2^{-1 + 4} = 2^3$ .

Thus, the first expression simplifies to $2^3$ .

For the second expression $2^3 \cdot 2^{-2} \cdot 2^5$ :

Apply the multiplication of powers rule: $2^3 \cdot 2^{-2} = 2^{3 + (-2)} = 2^1$ .
Now, multiply by $2^5$ : $2^1 \cdot 2^5 = 2^{1 + 5} = 2^6$ .

Thus, the second expression simplifies to $2^6$ .

To compare $2^3$ and $2^6$ , we recognize that $2^6$ is greater than $2^3$ . Hence, the second expression is greater.

Thus, the correct answer is: $<$ .

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