\frac{3^{15} \cdot 2^{34}}{3^{13} \cdot 2^{32}}
Answer (2)
To simplify the expression 3 13 ⋅ 2 32 3 15 ⋅ 2 34 , apply the exponent rule to get 36 . First, subtract the exponents for both bases, then calculate the powers and multiply the results. The final answer is 36 .
;
Apply the exponent rule a n a m = a m − n to simplify the expression.
Calculate 3 15 − 13 = 3 2 = 9 and 2 34 − 32 = 2 2 = 4 .
Multiply the results: 9 \t ⋅ 4 = 36 .
The simplified value of the expression is 36 .
Explanation
Understanding the Problem We are given the expression 3 13 \t ⋅ 2 32 3 15 \t ⋅ 2 34 . Our goal is to simplify it using exponent rules.
Applying Exponent Rules We will use the rule a n a m = a m − n . Applying this to the given expression, we get: 3 13 \t ⋅ 2 32 3 15 \t ⋅ 2 34 = 3 13 3 15 \t ⋅ 2 32 2 34 = 3 15 − 13 \t ⋅ 2 34 − 32 = 3 2 \t ⋅ 2 2
Calculating Powers Now we calculate the values of 3 2 and 2 2 :
3 2 = 3 \t ⋅ 3 = 9 2 2 = 2 \t ⋅ 2 = 4
Final Calculation Finally, we multiply the results: 3 2 \t ⋅ 2 2 = 9 \t ⋅ 4 = 36 So, the simplified expression is 36.
Examples
Understanding exponent rules is crucial in many scientific and engineering applications. For instance, in computer science, when dealing with data storage, we often encounter powers of 2. Simplifying expressions with exponents helps in calculating storage capacities and data transfer rates. Similarly, in physics, exponential functions are used to describe phenomena like radioactive decay, where simplifying exponential expressions can help determine the half-life of a substance or the amount of remaining material after a certain time.
