$\frac{3^{15} \cdot 2^{34}}{3^{33} \cdot 2^{32}}$
Answer (2)
To simplify the expression 3 33 ⋅ 2 32 3 15 ⋅ 2 34 , we rewrite it using the quotient rule, leading to 3 18 4 . The negative exponent converts to a fraction with a positive exponent. The final form is 3 18 4 .
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Rewrite the expression as a product of fractions with the same base: 3 33 3 15 ⋅ 2 32 2 34 .
Apply the quotient rule of exponents: 3 15 − 33 ⋅ 2 34 − 32 = 3 − 18 ⋅ 2 2 .
Rewrite the expression using positive exponents: 3 18 2 2 .
Simplify the expression: 3 18 4 .
Explanation
Understanding the Problem We are given the expression 3 33 ⋅ 2 32 3 15 ⋅ 2 34 and we want to simplify it.
Rewriting the Expression To simplify the expression, we will use the properties of exponents. Specifically, we will use the rule that a n a m = a m − n . We can rewrite the expression as a product of two fractions: 3 33 ⋅ 2 32 3 15 ⋅ 2 34 = 3 33 3 15 ⋅ 2 32 2 34
Applying the Quotient Rule Now, we apply the rule a n a m = a m − n to each fraction: 3 33 3 15 = 3 15 − 33 = 3 − 18 2 32 2 34 = 2 34 − 32 = 2 2
Simplifying Negative Exponent So, the expression becomes: 3 − 18 ⋅ 2 2 We also know that a − n = a n 1 , so we can rewrite 3 − 18 as 3 18 1 . Thus, the expression is: 3 18 1 ⋅ 2 2 = 3 18 2 2
Calculating the Final Value Since 2 2 = 4 , the simplified expression is: 3 18 4 We know that 3 18 = 387420489 . Therefore, the final simplified expression is: 387420489 4
Final Answer Therefore, the simplified form of the given expression is 3 18 4 .
Examples
Imagine you are comparing the processing power of two computers. One computer can perform 3 33 operations per second with 2 32 bits of data, while another can perform 3 15 operations per second with 2 34 bits of data. Simplifying the expression 3 33 ⋅ 2 32 3 15 ⋅ 2 34 helps you determine the relative performance ratio between these computers, showing how much more efficient one is compared to the other. This type of calculation is useful in various fields, such as engineering, computer science, and data analysis, where comparing exponential quantities is essential.
