Find the Domain ( fifth root of 5)^(3-x)=1/25

Problem

(√(5,5))(3−x)=1/25

Solution

1. Rewrite the radical expression using a fractional exponent.

(√(5,5))(3−x)=(5(1/5))(3−x)

2. Apply the power of a power rule (am)n=a(m⋅n) to the left side.

5=1/25

3. Express the right side as a power of 5 to create a common base.

1/25=1/5=5(−2)

4. Equate the exponents since the bases are now identical.

(3−x)/5=−2

5. Solve for x by multiplying both sides by 5.

3−x=−10

6. Isolate x by subtracting 3 from both sides and then multiplying by −1

−x=−13

x=13

7. Determine the domain of the original equation. Since the equation involves an exponential function with a constant base and a linear exponent, the expression is defined for all real numbers.

Domain=(−∞,∞)

Final Answer

Domain=(−∞,∞)

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