Use the power rules for exponents. [tex]$\left(\frac{5}{9}\right)^6 \cdot 5^2$[/tex]
Answer (2)
531441 390625
Explanation
Understanding the Problem We are given the expression ( 9 5 ) 6 ⋅ 5 2 . Our goal is to simplify this expression using the power rules of exponents.
Rewriting the Expression First, let's rewrite the expression as a fraction: ( 9 5 ) 6 ⋅ 5 2 = 9 6 5 6 ⋅ 5 2
Simplifying the Denominator Now, we can rewrite 9 as 3 2 , so the expression becomes: ( 3 2 ) 6 5 6 ⋅ 5 2 = 3 12 5 6 ⋅ 5 2
Combining Powers Next, we combine the powers of 5 in the numerator: 3 12 5 6 ⋅ 5 2 = 3 12 5 6 + 2 = 3 12 5 8
Calculating the Powers Now, let's calculate 5 8 and 3 12 : 5 8 = 390625 3 12 = 531441
Final Result So, the simplified expression is: 3 12 5 8 = 531441 390625 ≈ 0.735
Conclusion Therefore, the simplified form of the given expression is 531441 390625 .
Examples
Understanding and simplifying expressions with exponents is crucial in many fields, such as physics and engineering, where exponential growth and decay are common. For example, calculating the decay of radioactive materials or the growth of a bacterial population involves simplifying exponential expressions. This skill also helps in financial mathematics when dealing with compound interest calculations.
We simplified the expression ( 9 5 ) 6 ⋅ 5 2 to 3 12 5 8 , which equals 531441 390625 . The steps involved rewriting the expression, simplifying the denominator, and combining the powers of 5. The final answer is 531441 390625 .
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