Evaluate (5750 root of 0.5)^4000

Problem

(√(5750,0.5))4000

Solution

1. Rewrite the radical expression using a fractional exponent.

√(5750,0.5)=0.5

2. Apply the power of a power rule, which states that (am)n=a(m⋅n)

(0.5)4000=0.5

3. Simplify the fraction in the exponent by dividing both the numerator and denominator by their greatest common divisor, 250.

4000/5750=16/23

4. Substitute the simplified exponent back into the expression.

0.5

5. Convert the decimal base to a fraction for a more standard exact form.

(1/2)16/23

6. Express the result using radical notation if preferred.

√(23,(1/2)16)

Final Answer

(√(5750,0.5))4000=0.5

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