Compute: [tex]$\frac{0.6^{12}}{0.6^4 \cdot 0.6^5}$[/tex]
Answer (2)
Simplify the denominator using the exponent rule: 0. 6 4 c d o t 0. 6 5 = 0. 6 4 + 5 = 0. 6 9 .
Simplify the fraction using the exponent rule: 0. 6 9 0. 6 12 = 0. 6 12 − 9 = 0. 6 3 .
Calculate the final value: 0. 6 3 = 0.6 c d o t 0.6 c d o t 0.6 = 0.216 .
The final answer is 0.216 .
Explanation
Understanding the Problem We are asked to compute the value of the expression 0. 6 4 ⋅ 0. 6 5 0. 6 12 . This involves exponents and division. The base is the same in all terms, which is 0.6.
Simplifying the Denominator First, we simplify the denominator using the rule a m ⋅ a n = a m + n . So, 0. 6 4 ⋅ 0. 6 5 = 0. 6 4 + 5 = 0. 6 9 .
Rewriting the Expression Now the expression becomes 0. 6 9 0. 6 12 .
Simplifying the Fraction Next, we use the rule a n a m = a m − n . So, 0. 6 9 0. 6 12 = 0. 6 12 − 9 = 0. 6 3 .
Calculating the Final Value Finally, we compute 0. 6 3 = 0.6 ⋅ 0.6 ⋅ 0.6 = 0.36 ⋅ 0.6 = 0.216 .
Final Answer Therefore, the value of the expression is 0.216.
Examples
Understanding exponents and how to manipulate them is useful in many areas, such as calculating compound interest. For example, if you invest $100 at an annual interest rate of 6% compounded annually, the amount you have after 3 years is $100 \cdot (1.06)^3 = $100 \cdot 1.191016 = $119.10 (approximately). This is an example of exponential growth, and the rules of exponents are crucial for understanding and calculating such growth.
The expression 0. 6 4 ⋅ 0. 6 5 0. 6 12 simplifies to 0. 6 12 − 9 = 0. 6 3 , which equals 0.216 . Thus, the final answer is 0.216.
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