\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 using the properties of exponents to combine and divide the powers. First, we find the numerator as 2 12 and the denominator as 2 7 , resulting in 2 12 − 7 = 2 5 . The final value of 2 5 is 32.
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Multiply the powers in the numerator: 2 4 × 2 3 × 2 5 = 2 12 .
Multiply the powers in the denominator: 2 4 × 2 3 = 2 7 .
Divide the simplified numerator by the simplified denominator: 2 7 2 12 = 2 12 − 7 = 2 5 .
Calculate the final value: 2 5 = 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 combine and simplify the expression.
Simplifying the Numerator First, we simplify the numerator using the property a m × a n = a m + n . So, 2 4 × 2 3 × 2 5 = 2 4 + 3 + 5 = 2 12 .
Simplifying the Denominator Next, we simplify the denominator using the same property: 2 4 × 2 3 = 2 4 + 3 = 2 7 .
Dividing Powers with the Same Base Now, we rewrite the expression as 2 7 2 12 . To simplify this fraction, we use the property a n a m = a m − n . Therefore, 2 7 2 12 = 2 12 − 7 = 2 5 .
Calculating the Final Value Finally, we calculate 2 5 = 32 .
Examples
Understanding exponents is crucial in many fields, such as computer science when dealing with binary numbers and data storage, or in finance when calculating compound interest. For instance, if a bacteria doubles every hour, the number of bacteria after n hours can be modeled as 2 n . Simplifying expressions with exponents helps in predicting growth rates and making informed decisions.
