[tex]v^{\frac{5}{4}}=243[/tex]
Answer (2)
Raise both sides of the equation to the power of 5 4 : v = 24 3 5 4 .
Express 243 as 3 5 : v = ( 3 5 ) 5 4 .
Simplify the exponent: v = 3 5 ⋅ 5 4 = 3 4 .
Calculate 3 4 to find the value of v : 81 .
Explanation
Problem Analysis We are given the equation v 4 5 = 243 and we need to find the value of v .
Isolating v To isolate v , we raise both sides of the equation to the power of 5 4 . This gives us v = 24 3 5 4 .
Expressing 243 as a power of 3 We can express 243 as 3 5 , so we have v = ( 3 5 ) 5 4 .
Simplifying the exponent Now we simplify the exponent: v = 3 5 ⋅ 5 4 = 3 4 .
Calculating 3^4 Finally, we calculate 3 4 to find the value of v : v = 3 4 = 81 .
Final Answer Therefore, the value of v is 81.
Examples
Understanding exponential equations like this is crucial in various fields, such as calculating growth rates in biology, determining compound interest in finance, and modeling radioactive decay in physics. For instance, if a bacterial population grows according to the equation P = P 0 \t 5/4 , where P is the population at time t and P 0 is the initial population, solving for t when P is a certain multiple of P 0 involves similar algebraic manipulations.
To solve the equation v 4 5 = 243 , we raise both sides to the power of 5 4 , allowing us to simplify it to find that v = 81 . The process involves expressing 243 as a power of 3 and working through exponent rules. Ultimately, the solution shows that v = 81 .
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