b. $(7 \cdot 5)^{-1}=$ $\square$
Answer (1)
First, calculate the product of 7 and 5: 7 ⋅ 5 = 35 .
Then, find the reciprocal of the product: ( 7 ⋅ 5 ) − 1 = 3 5 − 1 .
Express the reciprocal as a fraction: 3 5 − 1 = 35 1 .
The final answer is 35 1 .
Explanation
Understanding the problem We are asked to evaluate the expression ( 7 c d o t 5 ) − 1 . This means we need to find the reciprocal of the product of 7 and 5.
Calculating the product First, we calculate the product of 7 and 5: 7 c d o t 5 = 35
Applying the exponent Next, we apply the exponent -1 to the result. Recall that a − 1 = a 1 . Therefore, we have: ( 7 c d o t 5 ) − 1 = 3 5 − 1 = 35 1
Final answer Thus, the final answer is 35 1 .
Examples
Understanding exponents and reciprocals is crucial in many areas, such as calculating interest rates or understanding exponential decay in science. For example, if you invest money at an annual interest rate, the reciprocal helps determine how long it takes for your investment to double. Similarly, in physics, understanding reciprocals is essential when dealing with inverse relationships, such as the relationship between resistance and conductance in electrical circuits. These concepts are fundamental in various fields and help in making informed decisions and predictions.
