\frac{2^4 \cdot 2^3 \cdot 2^3}{2^4 \cdot 2^3}
Answer (2)
To simplify the expression 2 4 ⋅ 2 3 2 4 ⋅ 2 3 ⋅ 2 3 , we first simplify the numerator to 2 10 and the denominator to 2 7 . Dividing these gives us 2 3 = 8 . Thus, the final answer is 8 .
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Simplify the numerator using the rule a m × a n = a m + n : 2 4 × 2 3 × 2 3 = 2 10 .
Simplify the denominator using the rule a m × a n = a m + n : 2 4 × 2 3 = 2 7 .
Simplify the fraction using the rule a n a m = a m − n : 2 7 2 10 = 2 3 .
Calculate the final value: 2 3 = 8 . The simplified expression is 8 .
Explanation
Understanding the problem We are asked to simplify the expression 2 4 × 2 3 2 4 × 2 3 × 2 3 . This involves using the properties of exponents to simplify the expression.
Simplifying the numerator First, let's simplify the numerator. We use the rule a m × a n = a m + n . So, 2 4 × 2 3 × 2 3 = 2 4 + 3 + 3 = 2 10 .
Simplifying the denominator Next, let's simplify the denominator. Using the same rule, 2 4 × 2 3 = 2 4 + 3 = 2 7 .
Simplifying the fraction Now, we simplify the entire expression. We use the rule a n a m = a m − n . So, 2 7 2 10 = 2 10 − 7 = 2 3 .
Calculating the final value Finally, we calculate 2 3 = 2 × 2 × 2 = 8 . Therefore, the simplified expression is 8.
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). It's also used in finance for calculating compound interest, where the initial investment grows exponentially over time. Moreover, exponents are fundamental in physics, particularly in describing phenomena like radioactive decay or exponential growth in population models. Mastering exponents provides a strong foundation for tackling complex problems in various scientific and practical contexts.
