Compute: [tex]$\frac{3^{15}}{3^6 \cdot 3^5}$[/tex]
Answer (1)
Simplify the denominator using the rule a m ⋅ a n = a m + n , resulting in 3 6 ⋅ 3 5 = 3 11 .
Simplify the expression using the rule a n a m = a m − n , resulting in 3 11 3 15 = 3 4 .
Compute the final value: 3 4 = 81 .
The value of the expression is 81 .
Explanation
Understanding the Problem We are asked to compute the value of the expression 3 6 ⋅ 3 5 3 15 . This involves exponents and division. Our goal is to simplify the expression using exponent rules and arrive at a numerical value.
Simplifying the Denominator First, we simplify the denominator using the rule a m ⋅ a n = a m + n . So, 3 6 ⋅ 3 5 = 3 6 + 5 = 3 11 .
Simplifying the Expression Now, the expression becomes 3 11 3 15 . We simplify this using the rule a n a m = a m − n . So, 3 11 3 15 = 3 15 − 11 = 3 4 .
Calculating the Final Value Finally, we compute 3 4 = 3 ⋅ 3 ⋅ 3 ⋅ 3 = 81 .
Conclusion Therefore, the value of the expression 3 6 ⋅ 3 5 3 15 is 81.
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) or in finance when calculating compound interest. For instance, if you invest $1000 at an annual interest rate of 5% compounded annually, the amount after 4 years can be calculated using exponents: 1000 × ( 1 + 0.05 ) 4 . Simplifying expressions with exponents helps in making quick estimations and understanding the scale of growth or decay in various real-world scenarios.
