\frac{2^4 \cdot 2^3 \cdot 2^5}{2^4 \cdot 2^3}
Answer (2)
The expression 2 4 ⋅ 2 3 2 4 ⋅ 2 3 ⋅ 2 5 simplifies to 32 by applying the properties of exponents. This involves simplifying both the numerator and the denominator and then reducing the fraction. The final answer is 32 .
;
Simplify the numerator using the property a m × a n = a m + n : 2 4 × 2 3 × 2 5 = 2 12 .
Simplify the denominator using the property a m × a n = a m + n : 2 4 × 2 3 = 2 7 .
Simplify the fraction using the property a n a m = a m − n : 2 7 2 12 = 2 5 .
Calculate the final value: 2 5 = 32 . The simplified expression is 32 .
Explanation
Understanding the Problem We are asked to simplify the expression 2 4 × 2 3 2 4 × 2 3 × 2 5 . This involves using the properties of exponents to simplify the numerator and the denominator, and then simplifying the entire fraction.
Simplifying the Numerator First, let's simplify the numerator. We use the property that a m × a n = a m + n . So, 2 4 × 2 3 × 2 5 = 2 4 + 3 + 5 = 2 12 .
Simplifying the Denominator Next, let's simplify the denominator. Again, we use the property that a m × a n = a m + n . So, 2 4 × 2 3 = 2 4 + 3 = 2 7 .
Simplifying the Fraction Now, we have the expression 2 7 2 12 . To simplify this fraction, we use the property that a n a m = a m − n . So, 2 7 2 12 = 2 12 − 7 = 2 5 .
Calculating the Final Value Finally, we calculate 2 5 = 32 . Therefore, the simplified expression is 32.
Examples
Exponents are used to model exponential growth and decay in various fields such as finance (compound interest), biology (population growth), and physics (radioactive decay). Simplifying expressions with exponents helps in understanding and predicting these phenomena. For example, calculating the future value of an investment with compound interest involves simplifying exponential expressions to determine the final amount. Similarly, in biology, exponential growth models are used to estimate population sizes over time, requiring simplification of exponential terms.
