\frac{2^4 \cdot 2^3 \cdot 2^5}{2^4 \cdot 2^3}
Answer (2)
To simplify the expression 2 4 ⋅ 2 3 2 4 ⋅ 2 3 ⋅ 2 5 , we first add the exponents in the numerator and denominator. This leads to 2 7 2 12 = 2 5 , which equals 32. Thus, the final answer is 32 .
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Simplify the numerator by adding the exponents: 2 4 \tcdot 2 3 \tcdot 2 5 = 2 12 .
Simplify the denominator by adding the exponents: 2 4 \tcdot 2 3 = 2 7 .
Divide the simplified numerator by the simplified denominator by subtracting the exponents: 2 7 2 12 = 2 5 .
Calculate the final result: 2 5 = 32 , so the answer is 32 .
Explanation
Understanding the Problem We are asked to simplify the expression 2 4 \tcdot 2 3 2 4 \tcdot 2 3 \tcdot 2 5 . This involves using the properties of exponents to combine and simplify the expression.
Simplifying the Numerator First, let's simplify the numerator. When multiplying exponential terms with the same base, we add the exponents: 2 4 \tcdot 2 3 \tcdot 2 5 = 2 4 + 3 + 5 = 2 12 .
Simplifying the Denominator Next, let's simplify the denominator. Similarly, 2 4 \tcdot 2 3 = 2 4 + 3 = 2 7 .
Dividing Numerator by Denominator Now, we divide the simplified numerator by the simplified denominator. When dividing exponential terms with the same base, we subtract the exponents: 2 7 2 12 = 2 12 − 7 = 2 5 .
Calculating the Final Result Finally, we calculate 2 5 = 2 \tcdot 2 \tcdot 2 \tcdot 2 \tcdot 2 = 32 . Therefore, the simplified expression is 32 .
Examples
Exponents are used to model exponential growth and decay in various real-world scenarios, such as population growth, compound interest, and radioactive decay. For example, if a population doubles every hour, the population size after t hours can be modeled as P ( t ) = P 0 \tcdot 2 t , where P 0 is the initial population. Simplifying expressions with exponents helps in making predictions and understanding the behavior of these phenomena.
