\frac{2^4 \cdot 2^5 \cdot 2^5}{2^3}=
Answer (2)
To simplify the expression 2 3 2 4 ⋅ 2 5 ⋅ 2 5 , we first combine the exponents in the numerator to get 2 14 and then simplify the division to find 2 11 , which equals 2048. Therefore, the final answer is 2048.
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Multiply the terms in the numerator using the exponent property: 2 4 × 2 5 × 2 5 = 2 4 + 5 + 5 = 2 14 .
Divide the result by the denominator using the exponent property: 2 3 2 14 = 2 14 − 3 = 2 11 .
Calculate the final value: 2 11 = 2048 .
The simplified expression is 2048 .
Explanation
Understanding the Problem We are asked to simplify the expression 2 3 2 4 × 2 5 × 2 5 . This involves using the properties of exponents to combine the terms in the numerator and then simplify the fraction.
Simplifying the Numerator First, we simplify the numerator using the property a m × a n = a m + n . So, 2 4 × 2 5 × 2 5 = 2 4 + 5 + 5 = 2 14 .
Simplifying the Fraction Now, we simplify the fraction using the property a n a m = a m − n . So, 2 3 2 14 = 2 14 − 3 = 2 11 .
Calculating the Final Value Finally, we calculate the value of 2 11 . 2 11 = 2048 .
Examples
Understanding exponents is crucial in many fields, such as computer science when dealing with memory sizes (e.g., kilobytes, megabytes, gigabytes, which are powers of 2). Also, exponents are used in finance to calculate compound interest, where the initial investment grows exponentially over time. For instance, if you invest P dollars at an annual interest rate r compounded n times per year for t years, the future value A of the investment is given by the formula A = P ( 1 + n r ) n t .
